Introduction to Rational Functions
0/44 completed

- Introduction
- Basic Case - Preface
- Three Basic Cases
- Basic Case I - Example
- Basic Case I - Recipe
- Basic Case II
- Basic Case III
- A Note on the Coefficient of X-Squared
- General Case - First Step
- General Case - Second Step
- General Case - Third Step
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24
- Exercise 25
- Exercise 26
- Exercise 27
- Exercise 28
- Exercise 29
- Exercise 30
- Exercise 31
- Exercise 32
- Exercise 33

{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Introduction to Rational Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"3m 44s","ChapterTopicVideoID":8326,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8326.jpeg","UploadDate":"2019-12-11T21:06:18.1370000","DurationForVideoObject":"PT3M44S","Description":null,"MetaTitle":"Introduction: Video + Workbook | Proprep","MetaDescription":"Integration of Rational Functions - Introduction to Rational Functions. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/integration-of-rational-functions/introduction-to-rational-functions/vid8497","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this clip, I\u0027m going to talk about integration of rational functions."},{"Start":"00:05.250 ","End":"00:09.480","Text":"I\u0027ll write the integral sign, something dx."},{"Start":"00:09.480 ","End":"00:11.910","Text":"Here I\u0027m going to put a rational function,"},{"Start":"00:11.910 ","End":"00:13.425","Text":"and I\u0027ll remind you what that is."},{"Start":"00:13.425 ","End":"00:15.915","Text":"It\u0027s a polynomial over a polynomial,"},{"Start":"00:15.915 ","End":"00:23.010","Text":"a_null plus a_1x plus a_2x squared,"},{"Start":"00:23.010 ","End":"00:24.120","Text":"plus and so on,"},{"Start":"00:24.120 ","End":"00:25.695","Text":"and so on, and so on,"},{"Start":"00:25.695 ","End":"00:32.640","Text":"up to a_nx to the n. On the denominator, something similar,"},{"Start":"00:32.640 ","End":"00:42.120","Text":"b _null, plus b_1x plus b_2x squared plus b_2x cubed, and so on,"},{"Start":"00:42.120 ","End":"00:49.070","Text":"and so on, up to b_mx to the m. Remember n is the degree of the polynomial,"},{"Start":"00:49.070 ","End":"00:51.020","Text":"and here m is the degree of the polynomial,"},{"Start":"00:51.020 ","End":"00:52.820","Text":"and they don\u0027t have to be the same degree."},{"Start":"00:52.820 ","End":"00:55.055","Text":"Also, some of the terms could be missing."},{"Start":"00:55.055 ","End":"00:58.025","Text":"If some a, say a_2 is 0,"},{"Start":"00:58.025 ","End":"01:00.380","Text":"then here we would have a missing term."},{"Start":"01:00.380 ","End":"01:02.090","Text":"Also, they don\u0027t have to be in order."},{"Start":"01:02.090 ","End":"01:04.820","Text":"Let\u0027s give an example of a rational function."},{"Start":"01:04.820 ","End":"01:09.610","Text":"Here we have an example of degree 4 over degree 2,"},{"Start":"01:09.610 ","End":"01:11.840","Text":"and a couple of things to notice."},{"Start":"01:11.840 ","End":"01:15.845","Text":"First of all, it\u0027s customary to actually write them in descending order of exponent,"},{"Start":"01:15.845 ","End":"01:18.605","Text":"as opposed to the way I presented it abstractly,"},{"Start":"01:18.605 ","End":"01:20.620","Text":"but you don\u0027t have to, it could be in any order."},{"Start":"01:20.620 ","End":"01:24.710","Text":"Secondly, it doesn\u0027t bother me that the x cubed term is missing."},{"Start":"01:24.710 ","End":"01:27.815","Text":"It\u0027s 0x cubed, so I just don\u0027t write it at all."},{"Start":"01:27.815 ","End":"01:34.740","Text":"Another example, here I\u0027ve got on the numerator a linear polynomial, it means degree 1,"},{"Start":"01:34.740 ","End":"01:36.940","Text":"and on the denominator degree 4,"},{"Start":"01:36.940 ","End":"01:40.670","Text":"and I\u0027m not concerned with the fact that x cubed is missing,"},{"Start":"01:40.670 ","End":"01:42.095","Text":"and x is missing."},{"Start":"01:42.095 ","End":"01:47.225","Text":"Another example, degree 2 polynomial over degree 2 polynomial."},{"Start":"01:47.225 ","End":"01:49.775","Text":"Let\u0027s keep on with some more examples."},{"Start":"01:49.775 ","End":"01:52.790","Text":"Here we have a degree 3 polynomial."},{"Start":"01:52.790 ","End":"01:54.860","Text":"On the denominator, it\u0027s a bit different."},{"Start":"01:54.860 ","End":"01:56.450","Text":"It\u0027s not in this form,"},{"Start":"01:56.450 ","End":"01:59.510","Text":"but if you multiply out, you square this,"},{"Start":"01:59.510 ","End":"02:01.505","Text":"and then multiply polynomials,"},{"Start":"02:01.505 ","End":"02:02.885","Text":"you\u0027ll get a polynomial."},{"Start":"02:02.885 ","End":"02:04.190","Text":"I even know what degree,"},{"Start":"02:04.190 ","End":"02:05.520","Text":"here it\u0027s degree 2,"},{"Start":"02:05.520 ","End":"02:06.950","Text":"and here after I square it,"},{"Start":"02:06.950 ","End":"02:09.380","Text":"it\u0027s degree 2, so altogether a degree 4 here,"},{"Start":"02:09.380 ","End":"02:11.485","Text":"and degree 3 in the numerator."},{"Start":"02:11.485 ","End":"02:15.650","Text":"Here\u0027s another example, in here too the denominator is"},{"Start":"02:15.650 ","End":"02:19.940","Text":"actually factorized and not given in this form, and that\u0027s okay."},{"Start":"02:19.940 ","End":"02:25.565","Text":"Another example, a degree 2 polynomial over a degree 2 polynomial,"},{"Start":"02:25.565 ","End":"02:27.905","Text":"integral of a rational function."},{"Start":"02:27.905 ","End":"02:30.350","Text":"I\u0027ll give you 3 more examples at once."},{"Start":"02:30.350 ","End":"02:31.640","Text":"In this 1 and this 1,"},{"Start":"02:31.640 ","End":"02:35.435","Text":"notice that we have a constant in the numerator."},{"Start":"02:35.435 ","End":"02:39.170","Text":"A constant fits the definition of a polynomial, because here,"},{"Start":"02:39.170 ","End":"02:44.555","Text":"if we take a_null to be equal to 1 or 4 as the case may be,"},{"Start":"02:44.555 ","End":"02:46.850","Text":"and all the rest of the constants are 0,"},{"Start":"02:46.850 ","End":"02:48.935","Text":"the rest of the coefficients I mean,"},{"Start":"02:48.935 ","End":"02:52.040","Text":"then we\u0027ll get a constant, and as usual,"},{"Start":"02:52.040 ","End":"02:55.445","Text":"polynomial can be an explicit form or it could be factored."},{"Start":"02:55.445 ","End":"02:57.755","Text":"Here we have a degree 10 polynomial,"},{"Start":"02:57.755 ","End":"03:01.920","Text":"and here we have a degree 10 plus 2 is 12 polynomial."},{"Start":"03:01.920 ","End":"03:05.135","Text":"In each case, polynomial over polynomial rational function,"},{"Start":"03:05.135 ","End":"03:10.220","Text":"and notice that a polynomial doesn\u0027t contain any square root signs,"},{"Start":"03:10.220 ","End":"03:11.720","Text":"or e to the power of,"},{"Start":"03:11.720 ","End":"03:13.190","Text":"or natural logarithm of,"},{"Start":"03:13.190 ","End":"03:15.484","Text":"or sine cosine, etc,"},{"Start":"03:15.484 ","End":"03:18.095","Text":"just polynomial over polynomial."},{"Start":"03:18.095 ","End":"03:23.300","Text":"Now, there is a very precise method of doing these integrals,"},{"Start":"03:23.300 ","End":"03:26.690","Text":"but it involves many cases and sub-cases."},{"Start":"03:26.690 ","End":"03:28.220","Text":"There\u0027s a lot of learning to do,"},{"Start":"03:28.220 ","End":"03:30.425","Text":"and a lot of different cases,"},{"Start":"03:30.425 ","End":"03:33.190","Text":"and in the rest of this clip and the following clips,"},{"Start":"03:33.190 ","End":"03:37.190","Text":"we\u0027ll be learning all the different kinds of rational functions,"},{"Start":"03:37.190 ","End":"03:39.980","Text":"and how to solve these integrals for all cases."},{"Start":"03:39.980 ","End":"03:45.720","Text":"This was just an intro to explain what rational functions are."}],"ID":8497},{"Watched":false,"Name":"Basic Case - Preface","Duration":"3m 41s","ChapterTopicVideoID":4479,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.640","Text":"Suppose we\u0027re asked to compute the integral of 4 over x minus 1 plus 10 over x minus 11."},{"Start":"00:08.640 ","End":"00:10.575","Text":"This is fairly straight forward."},{"Start":"00:10.575 ","End":"00:12.720","Text":"What we would do, very likely,"},{"Start":"00:12.720 ","End":"00:15.080","Text":"would be to split it up into the sum of"},{"Start":"00:15.080 ","End":"00:18.410","Text":"2 things and then take constants in front of the integral sign."},{"Start":"00:18.410 ","End":"00:24.050","Text":"We\u0027d end up with 4 times the integral of 1 over"},{"Start":"00:24.050 ","End":"00:31.070","Text":"x minus 1 plus 10 times the integral of 1 over x minus 11."},{"Start":"00:31.070 ","End":"00:34.250","Text":"Then what we would do would be to say, \"Okay,"},{"Start":"00:34.250 ","End":"00:41.465","Text":"the integral of 1 over x minus 1 is the natural log of absolute value of x minus 1,"},{"Start":"00:41.465 ","End":"00:44.360","Text":"either using a formula or to simply"},{"Start":"00:44.360 ","End":"00:47.765","Text":"noting that the numerator is the derivative of the denominator.\""},{"Start":"00:47.765 ","End":"00:49.670","Text":"Then we put the 4 in front,"},{"Start":"00:49.670 ","End":"00:51.650","Text":"plus, and similarly for this one,"},{"Start":"00:51.650 ","End":"00:57.470","Text":"this would be 10 times the natural log of x minus 11."},{"Start":"00:57.470 ","End":"00:59.585","Text":"Finally, we\u0027d add a constant."},{"Start":"00:59.585 ","End":"01:03.230","Text":"This is fairly straightforward and if you got it on the exam,"},{"Start":"01:03.230 ","End":"01:04.745","Text":"you\u0027d probably be delighted."},{"Start":"01:04.745 ","End":"01:06.740","Text":"But the professor thinks to himself,"},{"Start":"01:06.740 ","End":"01:08.480","Text":"\"Why don\u0027t I make it a bit harder?\""},{"Start":"01:08.480 ","End":"01:13.430","Text":"Then what he does is he decides to put these under a common denominator."},{"Start":"01:13.430 ","End":"01:18.340","Text":"He does a bit of algebra and he says something like 4 over x"},{"Start":"01:18.340 ","End":"01:24.785","Text":"minus 1 plus 10 over x minus 11 is equal to,"},{"Start":"01:24.785 ","End":"01:26.555","Text":"and I\u0027ll make a common denominator,"},{"Start":"01:26.555 ","End":"01:30.380","Text":"x minus 1, x minus 11."},{"Start":"01:30.380 ","End":"01:34.805","Text":"Then I\u0027ll do the usual by cross multiplication for example,"},{"Start":"01:34.805 ","End":"01:43.240","Text":"4 times x minus 11 plus 10 times x minus 1 over this."},{"Start":"01:43.240 ","End":"01:46.520","Text":"Then you open the brackets and collect xs together."},{"Start":"01:46.520 ","End":"01:50.525","Text":"So we get it from here, 4x plus 10x is 14x,"},{"Start":"01:50.525 ","End":"02:01.275","Text":"and then minus 44 minus 10 is minus 54 all this over x minus 1, x minus 11."},{"Start":"02:01.275 ","End":"02:07.820","Text":"Finally, what he does is to multiply the brackets on the denominator and we end up"},{"Start":"02:07.820 ","End":"02:16.940","Text":"with 14x minus 54 over x squared minus 12 x plus 11."},{"Start":"02:16.940 ","End":"02:22.220","Text":"Then on the exam you get the following integral to compute"},{"Start":"02:22.220 ","End":"02:30.365","Text":"14x minus 54 over x squared minus 12x plus 11."},{"Start":"02:30.365 ","End":"02:32.165","Text":"Now we don\u0027t really know how to do this."},{"Start":"02:32.165 ","End":"02:36.229","Text":"We wish that we had been given this exercise instead."},{"Start":"02:36.229 ","End":"02:39.994","Text":"What we need to do is the reverse process."},{"Start":"02:39.994 ","End":"02:44.875","Text":"Let me rephrase. What we know how to do is to take an algebraic expression like"},{"Start":"02:44.875 ","End":"02:50.670","Text":"4 over x minus 1 plus 10 over x minus 11."},{"Start":"02:50.670 ","End":"02:56.445","Text":"We know how to get from here to 14x minus"},{"Start":"02:56.445 ","End":"03:04.290","Text":"54 over x squared minus 12x plus 11."},{"Start":"03:04.290 ","End":"03:06.120","Text":"This is just what we did here,"},{"Start":"03:06.120 ","End":"03:08.495","Text":"very straightforward basic algebra."},{"Start":"03:08.495 ","End":"03:13.070","Text":"What we would like to do now is the reverse process is to get something like"},{"Start":"03:13.070 ","End":"03:19.160","Text":"this and be able to go the other way to take it from here and go to here."},{"Start":"03:19.160 ","End":"03:21.350","Text":"This is purely a question of algebra,"},{"Start":"03:21.350 ","End":"03:25.535","Text":"this is now not related to integrals all though we\u0027ll use it in integration."},{"Start":"03:25.535 ","End":"03:28.250","Text":"There is a technique of getting from here to here,"},{"Start":"03:28.250 ","End":"03:30.320","Text":"and it has a name,"},{"Start":"03:30.320 ","End":"03:33.050","Text":"decomposition into partial fraction."},{"Start":"03:33.050 ","End":"03:36.710","Text":"This is a major skill to learn if you want to do the"},{"Start":"03:36.710 ","End":"03:42.720","Text":"integral of rational functions and we\u0027ll see that in the following clips."}],"ID":4488},{"Watched":false,"Name":"Three Basic Cases","Duration":"6m 38s","ChapterTopicVideoID":8327,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"Continuing with the integration of rational functions,"},{"Start":"00:03.060 ","End":"00:05.775","Text":"this section is called the basic case."},{"Start":"00:05.775 ","End":"00:08.880","Text":"There is a certain basic integral of"},{"Start":"00:08.880 ","End":"00:14.610","Text":"a rational function and it\u0027s the key to solving just about any rational function."},{"Start":"00:14.610 ","End":"00:24.410","Text":"The basic integral, the mx plus n over x squared plus bx plus c,"},{"Start":"00:24.410 ","End":"00:26.115","Text":"the integral of this,"},{"Start":"00:26.115 ","End":"00:28.830","Text":"where m, n, b, and c are some constants."},{"Start":"00:28.830 ","End":"00:36.000","Text":"It turns out that this basic case has to be subdivided into 3 types of basic integral."},{"Start":"00:36.000 ","End":"00:37.440","Text":"Let\u0027s call them case 1, 2,"},{"Start":"00:37.440 ","End":"00:39.555","Text":"and 3, and let me present them."},{"Start":"00:39.555 ","End":"00:46.880","Text":"Case 1, the equation x squared plus bx plus c equals 0 has 2 different solutions."},{"Start":"00:46.880 ","End":"00:50.930","Text":"The denominator here is a quadratic and if I set it equal to 0,"},{"Start":"00:50.930 ","End":"00:56.070","Text":"it\u0027s a quadratic equation and a quadratic equation can either have 2 solutions,"},{"Start":"00:56.070 ","End":"00:58.985","Text":"1 solution, or no solutions."},{"Start":"00:58.985 ","End":"01:03.355","Text":"Anyway, it all revolves around this polynomial in the denominator."},{"Start":"01:03.355 ","End":"01:06.200","Text":"You could also, instead of talking about equations and solutions,"},{"Start":"01:06.200 ","End":"01:08.945","Text":"talk about number of roots of the polynomial."},{"Start":"01:08.945 ","End":"01:10.565","Text":"This either has 2 roots,"},{"Start":"01:10.565 ","End":"01:12.320","Text":"1 root or no roots,"},{"Start":"01:12.320 ","End":"01:14.375","Text":"and that will define our 3 cases."},{"Start":"01:14.375 ","End":"01:21.725","Text":"Case 1, the equation x squared plus bx plus c equals 0 has 2 different solutions."},{"Start":"01:21.725 ","End":"01:25.520","Text":"Another way of saying this is that this polynomial has 2 roots."},{"Start":"01:25.520 ","End":"01:29.810","Text":"Case 2 is when the equation has just 1 solution,"},{"Start":"01:29.810 ","End":"01:34.580","Text":"and the third case is when the equation has no solution."},{"Start":"01:34.580 ","End":"01:37.820","Text":"Those are the 3 cases that could be for any quadratic equation,"},{"Start":"01:37.820 ","End":"01:40.700","Text":"2 solutions, 1 solution, no solution."},{"Start":"01:40.700 ","End":"01:42.980","Text":"Now, I\u0027m going to write some examples of each."},{"Start":"01:42.980 ","End":"01:45.125","Text":"Here\u0027s 3 examples."},{"Start":"01:45.125 ","End":"01:49.804","Text":"Now each of them certainly fits the description of the basic integral,"},{"Start":"01:49.804 ","End":"01:53.360","Text":"which is generally a linear polynomial over a quadratic,"},{"Start":"01:53.360 ","End":"01:55.340","Text":"where the coefficient of x squared is 1."},{"Start":"01:55.340 ","End":"01:57.530","Text":"That\u0027s in words but in letters,"},{"Start":"01:57.530 ","End":"02:00.425","Text":"this is how it is and all of these fit except you might say"},{"Start":"02:00.425 ","End":"02:03.670","Text":"the last 1 because this doesn\u0027t look like mx plus n,"},{"Start":"02:03.670 ","End":"02:07.100","Text":"but really it is if you put m equals 0 and n equals 1,"},{"Start":"02:07.100 ","End":"02:08.600","Text":"then you won\u0027t get an x term."},{"Start":"02:08.600 ","End":"02:10.925","Text":"The numerator could also be something like 3x,"},{"Start":"02:10.925 ","End":"02:12.560","Text":"the n doesn\u0027t have to appear."},{"Start":"02:12.560 ","End":"02:14.690","Text":"These are all basic integrals."},{"Start":"02:14.690 ","End":"02:16.715","Text":"Now, why are they from case 1?"},{"Start":"02:16.715 ","End":"02:19.190","Text":"Well, if you just solve the quadratic equations,"},{"Start":"02:19.190 ","End":"02:25.970","Text":"I can tell you here that x equals 4 and x equals minus 1 will work here,"},{"Start":"02:25.970 ","End":"02:27.860","Text":"and here, x squared equals 1,"},{"Start":"02:27.860 ","End":"02:29.555","Text":"x is plus or minus 1."},{"Start":"02:29.555 ","End":"02:31.549","Text":"This has 2 roots also,"},{"Start":"02:31.549 ","End":"02:32.870","Text":"1 and minus 1."},{"Start":"02:32.870 ","End":"02:36.980","Text":"This, if you check it has 4 and 1 as its roots."},{"Start":"02:36.980 ","End":"02:39.290","Text":"The quadratic equation has 2 solutions."},{"Start":"02:39.290 ","End":"02:44.735","Text":"In this clip, I\u0027m just going to show you how to sort the basic integral into 3 types."},{"Start":"02:44.735 ","End":"02:46.910","Text":"Now let\u0027s get on to case 2,"},{"Start":"02:46.910 ","End":"02:48.955","Text":"and I\u0027ll give you also 3 examples."},{"Start":"02:48.955 ","End":"02:51.960","Text":"Here are the 3 examples."},{"Start":"02:51.960 ","End":"02:55.820","Text":"I actually took the same numerators as in case 1,"},{"Start":"02:55.820 ","End":"02:57.950","Text":"I just changed the denominators."},{"Start":"02:57.950 ","End":"03:01.325","Text":"Now the denominators are what we have to check and we want"},{"Start":"03:01.325 ","End":"03:05.225","Text":"to make sure that the solution to the equation, this equals 0."},{"Start":"03:05.225 ","End":"03:06.530","Text":"I mean there\u0027s only 1 of them."},{"Start":"03:06.530 ","End":"03:08.135","Text":"In this case, if you solve it,"},{"Start":"03:08.135 ","End":"03:11.465","Text":"only x equals minus 1 is a solution."},{"Start":"03:11.465 ","End":"03:17.030","Text":"This is what you get if you compute x plus 1 squared, I mean,"},{"Start":"03:17.030 ","End":"03:22.835","Text":"you must remember that there is this formula that a plus or minus b squared"},{"Start":"03:22.835 ","End":"03:29.610","Text":"is a squared plus or minus 2ab plus b squared."},{"Start":"03:29.610 ","End":"03:32.045","Text":"The solution to this is just minus 1."},{"Start":"03:32.045 ","End":"03:38.225","Text":"Similarly here, if you solve it the regular way you\u0027ll get only 1 solution, x equals 2,"},{"Start":"03:38.225 ","End":"03:44.725","Text":"or you might just be able to spot that this is x minus 2 all squared,"},{"Start":"03:44.725 ","End":"03:49.490","Text":"and here this solution is only minus 3 because"},{"Start":"03:49.490 ","End":"03:55.370","Text":"this factors into x plus 3 squared or you just solve it the way you know how to solve it."},{"Start":"03:55.370 ","End":"03:57.830","Text":"All these have exactly 1 solution."},{"Start":"03:57.830 ","End":"03:59.150","Text":"I\u0027ll just also write the solution."},{"Start":"03:59.150 ","End":"04:02.030","Text":"The solution here is minus 1."},{"Start":"04:02.030 ","End":"04:03.710","Text":"The solution here is 2,"},{"Start":"04:03.710 ","End":"04:05.785","Text":"the solution here is minus 3."},{"Start":"04:05.785 ","End":"04:08.475","Text":"Let\u0027s get on to case 3."},{"Start":"04:08.475 ","End":"04:12.275","Text":"Here\u0027s 3 examples that I claim apply to case 3,"},{"Start":"04:12.275 ","End":"04:16.310","Text":"where the denominator is never equal to 0, I claim."},{"Start":"04:16.310 ","End":"04:17.540","Text":"Let\u0027s see the first 1,"},{"Start":"04:17.540 ","End":"04:18.800","Text":"x squared plus 1."},{"Start":"04:18.800 ","End":"04:21.620","Text":"You can try and solve it and you\u0027ll see no solution."},{"Start":"04:21.620 ","End":"04:22.940","Text":"How do you tell there\u0027s no solution?"},{"Start":"04:22.940 ","End":"04:24.410","Text":"Because you get a minus number under"},{"Start":"04:24.410 ","End":"04:27.065","Text":"the square root sign if you use the quadratic formula."},{"Start":"04:27.065 ","End":"04:28.490","Text":"But you can also see it straight away,"},{"Start":"04:28.490 ","End":"04:31.040","Text":"x squared is at least 0 and plus 1,"},{"Start":"04:31.040 ","End":"04:34.850","Text":"it\u0027s strictly positive, so it certainly never 0."},{"Start":"04:34.850 ","End":"04:36.769","Text":"Wherever x is, it\u0027s positive."},{"Start":"04:36.769 ","End":"04:40.189","Text":"Similarly, this also is never 0."},{"Start":"04:40.189 ","End":"04:42.160","Text":"You can check the regular way."},{"Start":"04:42.160 ","End":"04:46.445","Text":"Here too the denominator is never equal to 0."},{"Start":"04:46.445 ","End":"04:49.040","Text":"In fact, in all of these cases,"},{"Start":"04:49.040 ","End":"04:50.255","Text":"there is a system,"},{"Start":"04:50.255 ","End":"04:52.325","Text":"and you can ignore this,"},{"Start":"04:52.325 ","End":"04:55.610","Text":"it\u0027s optional, but there is a way of telling fairly"},{"Start":"04:55.610 ","End":"04:59.135","Text":"quickly how many solutions a quadratic equation has."},{"Start":"04:59.135 ","End":"05:08.390","Text":"In general, quadratic equation is ax squared plus bx plus c equals 0 and a is not 0."},{"Start":"05:08.390 ","End":"05:11.270","Text":"Then we check something called the discriminant,"},{"Start":"05:11.270 ","End":"05:15.780","Text":"which is b squared minus 4ac,"},{"Start":"05:15.780 ","End":"05:18.365","Text":"and it can have 3 possibilities."},{"Start":"05:18.365 ","End":"05:22.010","Text":"It either comes out to be bigger than 0,"},{"Start":"05:22.010 ","End":"05:25.400","Text":"equals 0, or less than 0."},{"Start":"05:25.400 ","End":"05:27.590","Text":"If it comes out to be bigger than 0,"},{"Start":"05:27.590 ","End":"05:31.040","Text":"then we have 2 solutions to the quadratic,"},{"Start":"05:31.040 ","End":"05:33.560","Text":"and if it comes out equals to 0,"},{"Start":"05:33.560 ","End":"05:35.420","Text":"we only have 1 solution,"},{"Start":"05:35.420 ","End":"05:36.814","Text":"and if it\u0027s negative,"},{"Start":"05:36.814 ","End":"05:40.144","Text":"then there are no solutions or 0 solution."},{"Start":"05:40.144 ","End":"05:41.210","Text":"In fact, in our case,"},{"Start":"05:41.210 ","End":"05:43.940","Text":"it even comes out simpler because a is 1."},{"Start":"05:43.940 ","End":"05:49.790","Text":"Really, I only have to check b squared minus 4c because a here is just 1."},{"Start":"05:49.790 ","End":"05:52.290","Text":"It\u0027s a bit simpler, but this works."},{"Start":"05:52.290 ","End":"05:53.660","Text":"I\u0027ll just show you 1 example."},{"Start":"05:53.660 ","End":"05:55.445","Text":"If I take this case here,"},{"Start":"05:55.445 ","End":"06:02.230","Text":"b squared is 16 minus 4 times 1 times 4 is 0."},{"Start":"06:02.230 ","End":"06:04.635","Text":"Indeed it only has 1 solution."},{"Start":"06:04.635 ","End":"06:07.400","Text":"Here, I take b squared minus 4c,"},{"Start":"06:07.400 ","End":"06:12.620","Text":"so it\u0027s 4 squared minus 4 times 10 is 16 minus 40."},{"Start":"06:12.620 ","End":"06:15.425","Text":"It\u0027s negative, negative, no solution."},{"Start":"06:15.425 ","End":"06:17.075","Text":"This if you want to use it,"},{"Start":"06:17.075 ","End":"06:20.125","Text":"but the way you used to do it will work fine too."},{"Start":"06:20.125 ","End":"06:22.069","Text":"That\u0027s the 3 cases,"},{"Start":"06:22.069 ","End":"06:27.980","Text":"depending on the number of solutions that the denominator equals 0 gives."},{"Start":"06:27.980 ","End":"06:31.415","Text":"2 solutions, 1 solution, no solution,"},{"Start":"06:31.415 ","End":"06:34.040","Text":"and that makes it case 1, case 2,"},{"Start":"06:34.040 ","End":"06:39.600","Text":"or case 3, and we\u0027ll show you how to solve each case in its own right."}],"ID":8498},{"Watched":false,"Name":"Basic Case I - Example","Duration":"9m 45s","ChapterTopicVideoID":4480,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In the previous clip,"},{"Start":"00:02.430 ","End":"00:07.455","Text":"we learnt about the basic case of a rational function."},{"Start":"00:07.455 ","End":"00:11.325","Text":"I\u0027ll just write down to remind you what it was."},{"Start":"00:11.325 ","End":"00:16.110","Text":"It was the integral of mx plus n"},{"Start":"00:16.110 ","End":"00:22.200","Text":"over x squared plus bx plus c dx,"},{"Start":"00:22.200 ","End":"00:25.800","Text":"where m, n, b, and c are constants."},{"Start":"00:25.800 ","End":"00:30.915","Text":"We learned that there are 3 cases according to the denominator."},{"Start":"00:30.915 ","End":"00:34.380","Text":"If it has 2 roots, it\u0027s case 1,"},{"Start":"00:34.380 ","End":"00:37.080","Text":"if it has a single root, it\u0027s case 2,"},{"Start":"00:37.080 ","End":"00:40.625","Text":"and if it doesn\u0027t have any roots, then it\u0027s case 3."},{"Start":"00:40.625 ","End":"00:42.965","Text":"Now how do I know this is case 1?"},{"Start":"00:42.965 ","End":"00:45.860","Text":"If you check and you solve the equation,"},{"Start":"00:45.860 ","End":"00:52.589","Text":"that x squared minus 12x plus 11 equals zero,"},{"Start":"00:52.589 ","End":"00:54.900","Text":"that you indeed get 2 roots,"},{"Start":"00:54.900 ","End":"01:02.630","Text":"2 solutions and we get x equals 1 or x equals 11."},{"Start":"01:02.630 ","End":"01:04.760","Text":"It really is the case 1."},{"Start":"01:04.760 ","End":"01:11.325","Text":"In actual fact, we\u0027ve encountered this rational function in the introduction clip."},{"Start":"01:11.325 ","End":"01:19.345","Text":"We actually got to it by having 2 separate expressions in x,"},{"Start":"01:19.345 ","End":"01:23.095","Text":"simpler ones, and we combined them with a common denominator to get this."},{"Start":"01:23.095 ","End":"01:26.890","Text":"Remember there, I said that the clever thing now to do would be to go"},{"Start":"01:26.890 ","End":"01:31.495","Text":"backwards and from the combined to get to the 2 individual pieces."},{"Start":"01:31.495 ","End":"01:33.340","Text":"I\u0027m going to show you the technique."},{"Start":"01:33.340 ","End":"01:35.125","Text":"I\u0027ll show you how we do it in general."},{"Start":"01:35.125 ","End":"01:39.580","Text":"Now, when we have 2 roots of a quadratic polynomial,"},{"Start":"01:39.580 ","End":"01:41.665","Text":"then we can decompose it."},{"Start":"01:41.665 ","End":"01:49.330","Text":"What we say is that x squared minus 12x plus 11 is equal to,"},{"Start":"01:49.330 ","End":"01:56.395","Text":"we have x minus 1 root times x minus the other root."},{"Start":"01:56.395 ","End":"02:00.515","Text":"I could express it in general and tell you how it is in general,"},{"Start":"02:00.515 ","End":"02:03.590","Text":"if you take this equation equals 0,"},{"Start":"02:03.590 ","End":"02:06.275","Text":"or just say the roots of this polynomial,"},{"Start":"02:06.275 ","End":"02:10.355","Text":"and suppose they are x_1 and x_2,"},{"Start":"02:10.355 ","End":"02:13.320","Text":"like my 1 and 11 here."},{"Start":"02:14.330 ","End":"02:17.390","Text":"If you have the 2 roots,"},{"Start":"02:17.390 ","End":"02:21.815","Text":"then this thing is known to be equal to"},{"Start":"02:21.815 ","End":"02:27.985","Text":"x minus x_1 times x minus x_2."},{"Start":"02:27.985 ","End":"02:31.460","Text":"That\u0027s what the denominator is equal to."},{"Start":"02:31.460 ","End":"02:33.530","Text":"Where x_1 and x_2, as I say,"},{"Start":"02:33.530 ","End":"02:36.730","Text":"are these roots or the solutions of this equals 0."},{"Start":"02:36.730 ","End":"02:40.475","Text":"Here, we have 1 and 11 are the roots."},{"Start":"02:40.475 ","End":"02:46.850","Text":"So x minus 1 and x minus 11 is what this thing decomposes into."},{"Start":"02:46.850 ","End":"02:53.570","Text":"I can rewrite this rational function as 11x minus"},{"Start":"02:53.570 ","End":"03:00.530","Text":"54 over x minus"},{"Start":"03:00.530 ","End":"03:04.234","Text":"1 times x minus 11."},{"Start":"03:04.234 ","End":"03:09.350","Text":"Now, this we can be guaranteed that this thing will be of"},{"Start":"03:09.350 ","End":"03:19.325","Text":"the form A over x minus 1 plus B over x minus 11."},{"Start":"03:19.325 ","End":"03:22.835","Text":"We just have to figure out what are the constants A and B."},{"Start":"03:22.835 ","End":"03:27.350","Text":"Whenever we start out from this and the denominator factorizes,"},{"Start":"03:27.350 ","End":"03:31.640","Text":"theory of partial fractions says that it will equal this plus this,"},{"Start":"03:31.640 ","End":"03:35.135","Text":"and we just have to discover the missing constants."},{"Start":"03:35.135 ","End":"03:41.785","Text":"The way to do this is to multiply both sides by x minus 1, x minus 11."},{"Start":"03:41.785 ","End":"03:48.525","Text":"Here, I\u0027m left with 11x minus 54 multiplied by this."},{"Start":"03:48.525 ","End":"03:51.965","Text":"Here, if I multiply by the x minus 1 cancels."},{"Start":"03:51.965 ","End":"03:56.625","Text":"I\u0027m left with A times x minus 11."},{"Start":"03:56.625 ","End":"03:59.300","Text":"If I multiply these 2 by this,"},{"Start":"03:59.300 ","End":"04:05.900","Text":"x minus 11 cancels and I\u0027m left with B times x minus 1."},{"Start":"04:05.900 ","End":"04:09.830","Text":"The symbol equals is not really appropriate."},{"Start":"04:09.830 ","End":"04:13.100","Text":"We don\u0027t use much the symbol,"},{"Start":"04:13.100 ","End":"04:17.360","Text":"3 horizontal lines, which means identically equal to,"},{"Start":"04:17.360 ","End":"04:19.950","Text":"but it\u0027s appropriate here."},{"Start":"04:19.990 ","End":"04:25.850","Text":"This means that this is equal to this for all x that it\u0027s the same expression."},{"Start":"04:25.850 ","End":"04:29.180","Text":"It doesn\u0027t mean that we\u0027re solving for x or something."},{"Start":"04:29.180 ","End":"04:31.340","Text":"It means that whatever x we put in,"},{"Start":"04:31.340 ","End":"04:34.190","Text":"it\u0027s equal. That\u0027s an identity."},{"Start":"04:34.190 ","End":"04:39.395","Text":"Just like we have things like x minus 1,"},{"Start":"04:39.395 ","End":"04:43.520","Text":"x plus 1 equals x squared minus 1."},{"Start":"04:43.520 ","End":"04:45.035","Text":"It\u0027s not an equation,"},{"Start":"04:45.035 ","End":"04:49.010","Text":"it\u0027s an identity because it\u0027s true for all x."},{"Start":"04:49.010 ","End":"04:54.815","Text":"What I intend to do is substitute 2 different values of x."},{"Start":"04:54.815 ","End":"04:58.080","Text":"Then I\u0027ll get 2 equations and 2 unknowns,"},{"Start":"04:58.080 ","End":"05:00.285","Text":"A and B, and solve it."},{"Start":"05:00.285 ","End":"05:04.465","Text":"You could choose 2 different values of x at random,"},{"Start":"05:04.465 ","End":"05:08.300","Text":"and you would get A and B eventually."},{"Start":"05:08.300 ","End":"05:10.759","Text":"But to make things quicker,"},{"Start":"05:10.759 ","End":"05:14.930","Text":"it\u0027s easier to substitute particular values of x."},{"Start":"05:14.930 ","End":"05:18.245","Text":"For example, notice that if I put x equals 11 here,"},{"Start":"05:18.245 ","End":"05:22.445","Text":"this thing will be 0 and that\u0027ll help me and I\u0027ll be able to get B right away."},{"Start":"05:22.445 ","End":"05:25.190","Text":"I would rather not get into 2 equations and 2 unknowns."},{"Start":"05:25.190 ","End":"05:29.730","Text":"I\u0027ll rather have separately 2 equations each and 1 unknown,"},{"Start":"05:29.730 ","End":"05:31.560","Text":"1 in A and 1 in B."},{"Start":"05:31.560 ","End":"05:35.140","Text":"Let\u0027s substitute x equals 11."},{"Start":"05:36.220 ","End":"05:42.920","Text":"Oops, I just realized that I copied the original exercise wrongly."},{"Start":"05:42.920 ","End":"05:45.095","Text":"This was supposed to be 14x."},{"Start":"05:45.095 ","End":"05:49.630","Text":"This should be a 14, this should be a 14,"},{"Start":"05:49.630 ","End":"05:52.050","Text":"and this should be a 14."},{"Start":"05:52.050 ","End":"05:54.374","Text":"Let\u0027s just continue."},{"Start":"05:54.374 ","End":"05:58.575","Text":"What we get is 154,"},{"Start":"05:58.575 ","End":"06:05.385","Text":"14 times 11 minus 54 is 100."},{"Start":"06:05.385 ","End":"06:11.100","Text":"So a 100 equals 11 minus 11 is 0."},{"Start":"06:11.100 ","End":"06:13.185","Text":"That\u0027s what we wanted. We wanted A to disappear."},{"Start":"06:13.185 ","End":"06:15.600","Text":"11 minus 1 is 10,"},{"Start":"06:15.600 ","End":"06:19.320","Text":"so it\u0027s B10 or let me write it as 10B,"},{"Start":"06:19.320 ","End":"06:26.190","Text":"and therefore, we get that B equals 10, that\u0027s B."},{"Start":"06:26.190 ","End":"06:27.990","Text":"Now let\u0027s go for A."},{"Start":"06:27.990 ","End":"06:31.025","Text":"You might have guessed that to get A,"},{"Start":"06:31.025 ","End":"06:34.555","Text":"we just have to substitute x equals 1."},{"Start":"06:34.555 ","End":"06:37.855","Text":"That\u0027s the thing that makes B 0."},{"Start":"06:37.855 ","End":"06:44.600","Text":"Here we go. We substitute in this equation, x equals 1."},{"Start":"06:44.600 ","End":"06:50.420","Text":"This time we get 14 times 1 minus 54 is minus"},{"Start":"06:50.420 ","End":"06:57.465","Text":"40 equals 1 minus 11 is minus 10."},{"Start":"06:57.465 ","End":"07:00.405","Text":"A minus 10 means minus 10A,"},{"Start":"07:00.405 ","End":"07:05.400","Text":"minus 10A, and this is zero."},{"Start":"07:05.400 ","End":"07:12.780","Text":"This gives us, today is minus 40 over minus 10, which is 4."},{"Start":"07:12.780 ","End":"07:16.200","Text":"I have both A and B."},{"Start":"07:16.200 ","End":"07:20.310","Text":"What this means is that this thing equal to,"},{"Start":"07:20.310 ","End":"07:21.615","Text":"we\u0027ll use another color,"},{"Start":"07:21.615 ","End":"07:32.320","Text":"it\u0027s 4 over x minus 1 plus 10 over x minus 11."},{"Start":"07:32.320 ","End":"07:35.825","Text":"Just want to say again that we didn\u0027t have to substitute"},{"Start":"07:35.825 ","End":"07:40.775","Text":"necessarily x equals 11 and x equals 1."},{"Start":"07:40.775 ","End":"07:43.190","Text":"We could have taken any 2 values of x,"},{"Start":"07:43.190 ","End":"07:45.320","Text":"it just would have made it a bit longer."},{"Start":"07:45.320 ","End":"07:47.720","Text":"You would have just got 2 equations and 2 unknowns,"},{"Start":"07:47.720 ","End":"07:52.190","Text":"A and B and you\u0027d still get the same answer and you can try it."},{"Start":"07:52.190 ","End":"07:55.790","Text":"Now, the main thing that we\u0027ve shown here is that"},{"Start":"07:55.790 ","End":"08:03.800","Text":"this rational function which we were given originally is equal to this rational function."},{"Start":"08:03.800 ","End":"08:05.690","Text":"Well, it doesn\u0027t look like a rational function."},{"Start":"08:05.690 ","End":"08:09.365","Text":"It\u0027s a decomposed rational function in 2 bits that say,"},{"Start":"08:09.365 ","End":"08:11.575","Text":"this is equal to this."},{"Start":"08:11.575 ","End":"08:14.430","Text":"This is much easier to solve."},{"Start":"08:14.430 ","End":"08:21.590","Text":"It turns out that instead of computing our original integral that all we have to do"},{"Start":"08:21.590 ","End":"08:23.695","Text":"now is integrate"},{"Start":"08:23.695 ","End":"08:33.515","Text":"4 over x minus 1 plus 10 over x minus 11 dx."},{"Start":"08:33.515 ","End":"08:36.305","Text":"Like I said, we did this before in the introduction,"},{"Start":"08:36.305 ","End":"08:39.710","Text":"what we do is we separate it into 2 pieces,"},{"Start":"08:39.710 ","End":"08:41.210","Text":"this piece plus this piece,"},{"Start":"08:41.210 ","End":"08:44.600","Text":"because an integral of a sum is the sum of the integrals."},{"Start":"08:44.600 ","End":"08:47.600","Text":"But we also can take a constant outside the integral."},{"Start":"08:47.600 ","End":"08:56.135","Text":"We get 4 times the integral of 1 over x minus 1dx plus 10"},{"Start":"08:56.135 ","End":"08:59.850","Text":"times the integral of 1 over"},{"Start":"08:59.850 ","End":"09:06.740","Text":"x minus 11 dx."},{"Start":"09:06.740 ","End":"09:11.195","Text":"Continuing, this is equal to 4 times."},{"Start":"09:11.195 ","End":"09:18.555","Text":"Now the integral of 1 over x minus 1 is the natural log of x minus 1 an absolute value,"},{"Start":"09:18.555 ","End":"09:22.130","Text":"either by using the appropriate formula or just by"},{"Start":"09:22.130 ","End":"09:25.940","Text":"noticing that the numerator is the derivative of the denominator,"},{"Start":"09:25.940 ","End":"09:29.285","Text":"and so the answer is natural log of the denominator."},{"Start":"09:29.285 ","End":"09:34.520","Text":"Similarly here we get 10 natural log of"},{"Start":"09:34.520 ","End":"09:39.995","Text":"x minus 11 in absolute value and plus a constant."},{"Start":"09:39.995 ","End":"09:45.930","Text":"That\u0027s essentially all there is to it for Basic case 1."}],"ID":4491},{"Watched":false,"Name":"Basic Case I - Recipe","Duration":"9m 22s","ChapterTopicVideoID":4481,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"I\u0027d like to now write down the recipe for"},{"Start":"00:03.060 ","End":"00:06.810","Text":"integrating rational functions which are of basic case 1,"},{"Start":"00:06.810 ","End":"00:09.195","Text":"where the denominator has 2 roots,"},{"Start":"00:09.195 ","End":"00:10.890","Text":"let\u0027s call them x_1 and x_2."},{"Start":"00:10.890 ","End":"00:14.475","Text":"When I say recipe, I just mean a summary of the main steps."},{"Start":"00:14.475 ","End":"00:19.050","Text":"What we do is, we start out with the original exercise,"},{"Start":"00:19.050 ","End":"00:23.730","Text":"which is given in the form mx plus n over x"},{"Start":"00:23.730 ","End":"00:29.415","Text":"squared plus bx plus C. The integral of that is what we want,"},{"Start":"00:29.415 ","End":"00:30.765","Text":"that\u0027s the basic case,"},{"Start":"00:30.765 ","End":"00:33.570","Text":"and we\u0027re assuming it\u0027s of type 1,"},{"Start":"00:33.570 ","End":"00:36.085","Text":"which is 2 roots, x_1 and x_2."},{"Start":"00:36.085 ","End":"00:43.010","Text":"Using that, we can rewrite it as mx plus n over x minus x_1,"},{"Start":"00:43.010 ","End":"00:47.540","Text":"x minus x_2, x_1 and x_2 are these roots."},{"Start":"00:47.540 ","End":"00:51.499","Text":"After that, we use the theory of partial fractions,"},{"Start":"00:51.499 ","End":"00:57.500","Text":"like the example we did before to break this part up into A"},{"Start":"00:57.500 ","End":"01:04.620","Text":"over x minus x_1 plus B over x minus x_2,"},{"Start":"01:04.620 ","End":"01:06.770","Text":"where A and B are numbers."},{"Start":"01:06.770 ","End":"01:10.730","Text":"That\u0027s what we spent a lot of time doing just earlier on."},{"Start":"01:10.730 ","End":"01:13.040","Text":"Now that I have it in partial fractions,"},{"Start":"01:13.040 ","End":"01:15.845","Text":"the solution is almost immediate."},{"Start":"01:15.845 ","End":"01:20.650","Text":"A times natural log of x minus x_1,"},{"Start":"01:20.650 ","End":"01:27.985","Text":"plus B times natural log of x minus x_2 and plus as always,"},{"Start":"01:27.985 ","End":"01:30.495","Text":"the constant C of integration."},{"Start":"01:30.495 ","End":"01:33.065","Text":"This is an outline of the main steps that you follow."},{"Start":"01:33.065 ","End":"01:36.140","Text":"The hard part is the partial fractions,"},{"Start":"01:36.140 ","End":"01:38.600","Text":"to get from here to here to find these A and B,"},{"Start":"01:38.600 ","End":"01:40.420","Text":"but not very difficult really."},{"Start":"01:40.420 ","End":"01:41.970","Text":"Now that we\u0027ve got the recipe,"},{"Start":"01:41.970 ","End":"01:43.890","Text":"let\u0027s do another example."},{"Start":"01:43.890 ","End":"01:52.780","Text":"Let\u0027s take as our example the integral of 1 over x squared minus 1 dx."},{"Start":"01:52.780 ","End":"01:56.645","Text":"The first thing we do is factorize the denominator."},{"Start":"01:56.645 ","End":"01:58.850","Text":"Now here, of course there\u0027s a shortcut."},{"Start":"01:58.850 ","End":"02:02.005","Text":"We could use the difference of squares formula where"},{"Start":"02:02.005 ","End":"02:06.390","Text":"A squared minus B squared is equal to A minus B,"},{"Start":"02:06.390 ","End":"02:11.265","Text":"A plus B, but I don\u0027t want to do that because I want to stay more general."},{"Start":"02:11.265 ","End":"02:15.740","Text":"What usually do is solve the equation where x"},{"Start":"02:15.740 ","End":"02:19.850","Text":"squared minus 1 equals 0 and we\u0027ll get 2 solutions."},{"Start":"02:19.850 ","End":"02:24.485","Text":"x_1 will be minus 1 and x_2 will equal plus 1."},{"Start":"02:24.485 ","End":"02:25.880","Text":"I\u0027m not going to solve it as I said,"},{"Start":"02:25.880 ","End":"02:28.205","Text":"but you can always check that this works."},{"Start":"02:28.205 ","End":"02:33.050","Text":"Minus 1 squared minus 1 is 0 and 1 squared minus 1 is also 0."},{"Start":"02:33.050 ","End":"02:35.405","Text":"Now that we have the routes,"},{"Start":"02:35.405 ","End":"02:40.730","Text":"we can factorize the denominator as x minus one of the roots,"},{"Start":"02:40.730 ","End":"02:42.425","Text":"x minus the other root."},{"Start":"02:42.425 ","End":"02:45.950","Text":"Now we get the integral of 1 over,"},{"Start":"02:45.950 ","End":"02:49.200","Text":"I\u0027ll write it first as x minus x_1,"},{"Start":"02:49.200 ","End":"02:54.090","Text":"x minus x_2, and this is what I get after I replace x_1,"},{"Start":"02:54.090 ","End":"02:56.490","Text":"x_2 by minus 1 and 1."},{"Start":"02:56.490 ","End":"03:00.980","Text":"Now we\u0027ll use the method of partial fractions to decompose this."},{"Start":"03:00.980 ","End":"03:05.520","Text":"What we do is we write it as A over x"},{"Start":"03:05.520 ","End":"03:13.395","Text":"minus 1 plus B over x plus 1 and we find the constants A and B that make this work."},{"Start":"03:13.395 ","End":"03:15.290","Text":"It\u0027s like a reverse exercise."},{"Start":"03:15.290 ","End":"03:17.660","Text":"Normally we start off with 2 things like this and then we"},{"Start":"03:17.660 ","End":"03:20.360","Text":"put a common denominator and multiply this by this,"},{"Start":"03:20.360 ","End":"03:25.415","Text":"this plus this, and we end up with either a constant or something like mx plus n,"},{"Start":"03:25.415 ","End":"03:27.080","Text":"this time we\u0027re doing it in reverse."},{"Start":"03:27.080 ","End":"03:30.220","Text":"The technique is to multiply both sides"},{"Start":"03:30.220 ","End":"03:35.210","Text":"by the denominator here and multiply by x plus 1, x minus 1."},{"Start":"03:35.210 ","End":"03:37.670","Text":"Here I get just 1,"},{"Start":"03:37.670 ","End":"03:40.280","Text":"and here if I multiply this whole denominator,"},{"Start":"03:40.280 ","End":"03:41.870","Text":"the x minus 1 cancel,"},{"Start":"03:41.870 ","End":"03:44.615","Text":"so I get A times x plus 1."},{"Start":"03:44.615 ","End":"03:46.595","Text":"Here the x plus 1 cancels,"},{"Start":"03:46.595 ","End":"03:50.210","Text":"and I get B times x minus 1."},{"Start":"03:50.210 ","End":"03:52.400","Text":"Now remember how we do this."},{"Start":"03:52.400 ","End":"03:54.770","Text":"We can substitute any 2 numbers for x,"},{"Start":"03:54.770 ","End":"04:01.490","Text":"but the easiest is to substitute x equals minus 1 or x equals 1."},{"Start":"04:01.490 ","End":"04:04.220","Text":"If we let x equals 1,"},{"Start":"04:04.220 ","End":"04:07.210","Text":"we get that 1 equals,"},{"Start":"04:07.210 ","End":"04:09.525","Text":"I\u0027m just substituting x equals 1 here,"},{"Start":"04:09.525 ","End":"04:12.060","Text":"A times 1 plus 1 is 2,"},{"Start":"04:12.060 ","End":"04:13.860","Text":"so I write that as 2A,"},{"Start":"04:13.860 ","End":"04:17.960","Text":"and if I substitute x equals minus 1,"},{"Start":"04:17.960 ","End":"04:23.150","Text":"then 1 equals minus 1 plus 1 is nothing."},{"Start":"04:23.150 ","End":"04:26.975","Text":"This disappears, and minus 1 minus 1 is minus 2,"},{"Start":"04:26.975 ","End":"04:29.665","Text":"so I get minus 2B."},{"Start":"04:29.665 ","End":"04:33.530","Text":"This means that A equals a 1/2,"},{"Start":"04:33.530 ","End":"04:37.100","Text":"and this means that B equals minus a 1/2,"},{"Start":"04:37.100 ","End":"04:41.030","Text":"and that\u0027s the solution for this partial fraction."},{"Start":"04:41.030 ","End":"04:42.640","Text":"Now that I know what A and B are,"},{"Start":"04:42.640 ","End":"04:46.235","Text":"I can replace this expression with this expression."},{"Start":"04:46.235 ","End":"04:50.240","Text":"We get the integral of now A is a 1/2,"},{"Start":"04:50.240 ","End":"04:59.330","Text":"so it\u0027s a 1/2 over x minus 1 plus minus a 1/2 over x plus 1 dx."},{"Start":"04:59.330 ","End":"05:00.995","Text":"This we know how to do,"},{"Start":"05:00.995 ","End":"05:06.050","Text":"this is equal to 1/2 of natural logarithm of x"},{"Start":"05:06.050 ","End":"05:12.710","Text":"minus 1 minus a 1/2 times the log of x plus 1,"},{"Start":"05:12.710 ","End":"05:17.645","Text":"these are in absolute value and the constant of integration."},{"Start":"05:17.645 ","End":"05:20.270","Text":"It\u0027s possible to simplify this a bit."},{"Start":"05:20.270 ","End":"05:24.050","Text":"I can take the 1/2 outside brackets"},{"Start":"05:24.050 ","End":"05:28.340","Text":"and then I can see that I have a logarithm minus a logarithm."},{"Start":"05:28.340 ","End":"05:32.510","Text":"Subtracting logarithms is like dividing the argument of the log."},{"Start":"05:32.510 ","End":"05:36.170","Text":"We get natural log of this over this."},{"Start":"05:36.170 ","End":"05:39.110","Text":"I can even put it in 1 absolute value sign,"},{"Start":"05:39.110 ","End":"05:44.440","Text":"x minus 1 over x plus 1 and still plus the constant."},{"Start":"05:44.440 ","End":"05:47.060","Text":"If for some reason you\u0027re having difficulty with"},{"Start":"05:47.060 ","End":"05:50.060","Text":"the properties of the natural logarithm and you don\u0027t quite see this,"},{"Start":"05:50.060 ","End":"05:52.360","Text":"you could leave this as your final answer,"},{"Start":"05:52.360 ","End":"05:54.890","Text":"it\u0027s just customary to do it like this."},{"Start":"05:54.890 ","End":"05:59.340","Text":"A final example, very similar to the previous one."},{"Start":"05:59.340 ","End":"06:01.905","Text":"In fact, it\u0027s some generalization."},{"Start":"06:01.905 ","End":"06:10.684","Text":"What I want is the integral of 1 over x squared minus a squared dx."},{"Start":"06:10.684 ","End":"06:13.880","Text":"Previously, we had x squared minus 1,"},{"Start":"06:13.880 ","End":"06:16.370","Text":"so it fits the case where a equals 1."},{"Start":"06:16.370 ","End":"06:18.020","Text":"We\u0027ll generalize it a bit."},{"Start":"06:18.020 ","End":"06:21.215","Text":"Because it\u0027s similar, I\u0027ll go over it quicker."},{"Start":"06:21.215 ","End":"06:25.025","Text":"Once again, we factorize the denominator,"},{"Start":"06:25.025 ","End":"06:28.130","Text":"so we get that x squared minus a squared,"},{"Start":"06:28.130 ","End":"06:29.495","Text":"if we factorize it,"},{"Start":"06:29.495 ","End":"06:34.100","Text":"is x minus a times x plus a,"},{"Start":"06:34.100 ","End":"06:40.570","Text":"so that gives us the integral of 1 over x minus a,"},{"Start":"06:40.570 ","End":"06:43.290","Text":"x plus a dx."},{"Start":"06:43.290 ","End":"06:46.100","Text":"Now we use the partial fractions,"},{"Start":"06:46.100 ","End":"06:50.315","Text":"where I want to write 1 over x minus a,"},{"Start":"06:50.315 ","End":"06:54.660","Text":"x plus a, as A over x"},{"Start":"06:54.660 ","End":"07:01.470","Text":"minus 1 plus B over x plus a. I meant x minus a."},{"Start":"07:01.470 ","End":"07:03.530","Text":"Using the partial fractions,"},{"Start":"07:03.530 ","End":"07:08.055","Text":"we multiply out, so we multiply by this thing."},{"Start":"07:08.055 ","End":"07:10.065","Text":"This term becomes just 1,"},{"Start":"07:10.065 ","End":"07:13.385","Text":"this becomes A times x plus a,"},{"Start":"07:13.385 ","End":"07:17.955","Text":"and this becomes B times x minus a."},{"Start":"07:17.955 ","End":"07:21.900","Text":"Then we substitute 2 values for x,"},{"Start":"07:21.900 ","End":"07:27.170","Text":"any 2 values, but we want to make it simpler so I\u0027ll try a and minus a."},{"Start":"07:27.170 ","End":"07:29.620","Text":"If we substitute x equals a,"},{"Start":"07:29.620 ","End":"07:36.765","Text":"then this disappears and we get 1 equals A times a plus a."},{"Start":"07:36.765 ","End":"07:39.920","Text":"If we substitute x equals minus a,"},{"Start":"07:39.920 ","End":"07:47.615","Text":"then this disappears and we get 1 equals B times minus a minus a."},{"Start":"07:47.615 ","End":"07:50.735","Text":"In short, if we divide here by 2a,"},{"Start":"07:50.735 ","End":"07:55.235","Text":"we get that A equals 1 over 2a,"},{"Start":"07:55.235 ","End":"08:03.265","Text":"and here we get B equals 1 over minus 2a or minus 1 over 2a."},{"Start":"08:03.265 ","End":"08:06.950","Text":"We can then continue with this by just rewriting this in"},{"Start":"08:06.950 ","End":"08:10.985","Text":"the partial fraction form of a over x minus a,"},{"Start":"08:10.985 ","End":"08:16.965","Text":"which is 1 over 2a over x minus a,"},{"Start":"08:16.965 ","End":"08:18.600","Text":"and then here we have a minus,"},{"Start":"08:18.600 ","End":"08:26.435","Text":"so we write minus the integral of 1 over 2a over x plus a, all this dx,"},{"Start":"08:26.435 ","End":"08:35.480","Text":"and then this equals 1 over 2a natural log of the x minus a minus 1 over 2a,"},{"Start":"08:35.480 ","End":"08:37.685","Text":"it should be plus a here,"},{"Start":"08:37.685 ","End":"08:42.350","Text":"natural log of x plus a plus a constant."},{"Start":"08:42.350 ","End":"08:45.335","Text":"Just like before, we can simplify this,"},{"Start":"08:45.335 ","End":"08:48.050","Text":"you could leave it like this, that\u0027s an answer,"},{"Start":"08:48.050 ","End":"08:52.775","Text":"or we could simplify it by saying that this equals 1 over 2a,"},{"Start":"08:52.775 ","End":"08:55.870","Text":"the difference of the logs is the log of the quotient,"},{"Start":"08:55.870 ","End":"09:03.965","Text":"so we have natural log of x minus a over x plus a plus the constant."},{"Start":"09:03.965 ","End":"09:10.420","Text":"If a equals 1, notice that we get exactly the previous result from the previous example."},{"Start":"09:10.420 ","End":"09:13.400","Text":"Also like before, this last step is"},{"Start":"09:13.400 ","End":"09:17.420","Text":"optional if you\u0027re awkward with natural logarithms and manipulating them,"},{"Start":"09:17.420 ","End":"09:19.045","Text":"you can just leave it like this."},{"Start":"09:19.045 ","End":"09:23.910","Text":"That\u0027s the end of this example and of the whole clip."}],"ID":4490},{"Watched":false,"Name":"Basic Case II","Duration":"21m 28s","ChapterTopicVideoID":4483,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.975","Text":"We\u0027re continuing with the integration of rational functions."},{"Start":"00:03.975 ","End":"00:06.260","Text":"Last time we did Basic Case 1,"},{"Start":"00:06.260 ","End":"00:08.210","Text":"this time Basic Case 2."},{"Start":"00:08.210 ","End":"00:18.195","Text":"Perhaps I\u0027ll give you a quick reminder that the Basic Case was mx plus n"},{"Start":"00:18.195 ","End":"00:24.690","Text":"over x squared plus bx plus c. We"},{"Start":"00:24.690 ","End":"00:31.335","Text":"sorted it out into cases according to the number of roots that the denominator had."},{"Start":"00:31.335 ","End":"00:33.660","Text":"If it had 2 routes,"},{"Start":"00:33.660 ","End":"00:36.105","Text":"that gave us case 1."},{"Start":"00:36.105 ","End":"00:39.134","Text":"If it had 1 route only,"},{"Start":"00:39.134 ","End":"00:41.025","Text":"that\u0027s what we\u0027re going to do today,"},{"Start":"00:41.025 ","End":"00:44.025","Text":"this will be case 2."},{"Start":"00:44.025 ","End":"00:46.745","Text":"If it has 0 roots,"},{"Start":"00:46.745 ","End":"00:48.095","Text":"no roots at all,"},{"Start":"00:48.095 ","End":"00:51.705","Text":"that will make it case 3."},{"Start":"00:51.705 ","End":"00:53.970","Text":"This we did last time, this we\u0027ll do today."},{"Start":"00:53.970 ","End":"00:56.420","Text":"Let\u0027s start straight away with an example."},{"Start":"00:56.420 ","End":"01:01.205","Text":"We\u0027ll take the integral of x plus"},{"Start":"01:01.205 ","End":"01:08.980","Text":"1 over x squared minus 2x plus 1dx."},{"Start":"01:09.350 ","End":"01:12.115","Text":"How do I know it\u0027s case 2?"},{"Start":"01:12.115 ","End":"01:15.425","Text":"Well, I tested it out before I brought it."},{"Start":"01:15.425 ","End":"01:16.880","Text":"But when you get something like this,"},{"Start":"01:16.880 ","End":"01:18.830","Text":"you won\u0027t know which of the 3 cases it is."},{"Start":"01:18.830 ","End":"01:20.810","Text":"You\u0027ll see that it\u0027s a basic case."},{"Start":"01:20.810 ","End":"01:24.200","Text":"What you do is you just check how many roots this has."},{"Start":"01:24.200 ","End":"01:26.345","Text":"I\u0027m not going to do that work here."},{"Start":"01:26.345 ","End":"01:28.775","Text":"It turns out that there\u0027s only 1 single root,"},{"Start":"01:28.775 ","End":"01:30.985","Text":"and that is x equals 1."},{"Start":"01:30.985 ","End":"01:33.300","Text":"X equals 1 is the 1 root,"},{"Start":"01:33.300 ","End":"01:36.400","Text":"so we\u0027re in the case 2."},{"Start":"01:36.500 ","End":"01:43.510","Text":"Therefore that means that we can write the denominator as,"},{"Start":"01:43.510 ","End":"01:45.320","Text":"when it\u0027s single root,"},{"Start":"01:45.320 ","End":"01:48.470","Text":"it\u0027s always x minus that squared."},{"Start":"01:48.470 ","End":"01:58.184","Text":"We have integral of x plus 1 over x minus 1 squared dx."},{"Start":"01:58.184 ","End":"02:02.345","Text":"Now, of course, many of you would have seen these perfect squares."},{"Start":"02:02.345 ","End":"02:04.580","Text":"Probably you\u0027ve seen this 1 enough times already"},{"Start":"02:04.580 ","End":"02:06.710","Text":"to know straight away that it\u0027s a perfect square,"},{"Start":"02:06.710 ","End":"02:11.870","Text":"and it\u0027s x minus 1 squared and means that it only has a single root."},{"Start":"02:11.870 ","End":"02:14.000","Text":"But either way, you can just do it the long way,"},{"Start":"02:14.000 ","End":"02:16.505","Text":"find the solutions and only 1."},{"Start":"02:16.505 ","End":"02:20.425","Text":"Now, this is different from case 1."},{"Start":"02:20.425 ","End":"02:23.175","Text":"In case 1 we had 2 roots,"},{"Start":"02:23.175 ","End":"02:27.195","Text":"like say we had x minus 2x minus 3."},{"Start":"02:27.195 ","End":"02:31.100","Text":"Then we would try the method of partial fractions to see if"},{"Start":"02:31.100 ","End":"02:35.450","Text":"we could get the rational function in a certain format."},{"Start":"02:35.450 ","End":"02:37.190","Text":"I\u0027ll do it at the side."},{"Start":"02:37.190 ","End":"02:44.385","Text":"For example, if we had something over x minus 2,"},{"Start":"02:44.385 ","End":"02:49.985","Text":"x minus 3 doesn\u0027t quite matter what 2x minus 5."},{"Start":"02:49.985 ","End":"02:52.745","Text":"Then there is a certain format we look for,"},{"Start":"02:52.745 ","End":"02:59.820","Text":"and that was a over x minus 2 plus b over x minus 3."},{"Start":"02:59.820 ","End":"03:02.820","Text":"But here the 2 roots are 1 and 1."},{"Start":"03:02.820 ","End":"03:06.140","Text":"The question is, are we going to look for something of"},{"Start":"03:06.140 ","End":"03:12.620","Text":"the type a over x minus 1 plus b over x minus 1."},{"Start":"03:12.620 ","End":"03:16.850","Text":"The truth is that it\u0027s silly to do that."},{"Start":"03:16.850 ","End":"03:21.800","Text":"First of all, because if you have 2 fractions with the same denominator,"},{"Start":"03:21.800 ","End":"03:23.525","Text":"you can just add the numerators."},{"Start":"03:23.525 ","End":"03:25.520","Text":"It\u0027s as if I only had 1 of these."},{"Start":"03:25.520 ","End":"03:26.780","Text":"Let\u0027s use a different letter,"},{"Start":"03:26.780 ","End":"03:29.570","Text":"that\u0027s just equal to c over x minus 1."},{"Start":"03:29.570 ","End":"03:32.020","Text":"If C is A plus B,"},{"Start":"03:32.020 ","End":"03:33.530","Text":"if I add A plus B,"},{"Start":"03:33.530 ","End":"03:36.010","Text":"then I get just something over x minus 1."},{"Start":"03:36.010 ","End":"03:41.680","Text":"The question is, can something over x minus 1 squared equal x minus 1?"},{"Start":"03:41.680 ","End":"03:43.790","Text":"Well, it turns out that not,"},{"Start":"03:43.790 ","End":"03:46.550","Text":"but let\u0027s do it more rigorously."},{"Start":"03:46.550 ","End":"03:55.900","Text":"Suppose that I really had that x plus 1 over x minus 1 squared was equal."},{"Start":"03:55.900 ","End":"03:57.670","Text":"Again, when I say equal,"},{"Start":"03:57.670 ","End":"04:01.570","Text":"I mean identical to occasionally like to stress that it\u0027s not an equation in x."},{"Start":"04:01.570 ","End":"04:04.720","Text":"It\u0027s something that holds true for all x, that\u0027s an identity."},{"Start":"04:04.720 ","End":"04:09.540","Text":"Suppose this was equal to C over x minus 1,"},{"Start":"04:09.540 ","End":"04:11.475","Text":"where C is the A plus B."},{"Start":"04:11.475 ","End":"04:16.030","Text":"Then I would put both over the same common denominator,"},{"Start":"04:16.030 ","End":"04:18.220","Text":"which would be x minus 1 squared."},{"Start":"04:18.220 ","End":"04:22.090","Text":"We\u0027d get from here that x plus 1,"},{"Start":"04:22.090 ","End":"04:25.399","Text":"I\u0027m multiplying both sides by,"},{"Start":"04:25.490 ","End":"04:31.220","Text":"this I\u0027m leaving the same and this I\u0027m multiplying top and bottom by x minus 1."},{"Start":"04:31.220 ","End":"04:34.710","Text":"I\u0027d get C times x minus 1,"},{"Start":"04:34.710 ","End":"04:37.115","Text":"and get rid of the denominators."},{"Start":"04:37.115 ","End":"04:38.930","Text":"Now, since this is an identity,"},{"Start":"04:38.930 ","End":"04:41.320","Text":"and again, I\u0027ll put a third line in here."},{"Start":"04:41.320 ","End":"04:43.865","Text":"If it\u0027s true for every x,"},{"Start":"04:43.865 ","End":"04:47.930","Text":"then why don\u0027t I try substituting any value,"},{"Start":"04:47.930 ","End":"04:52.745","Text":"but I think I\u0027d like to take x equals 1 to make this thing 0."},{"Start":"04:52.745 ","End":"04:59.050","Text":"I will put x equals 1 in here,"},{"Start":"04:59.050 ","End":"05:03.709","Text":"and we get 1 plus 1 is 2,"},{"Start":"05:03.709 ","End":"05:05.690","Text":"and 1 minus 1 is 0,"},{"Start":"05:05.690 ","End":"05:08.110","Text":"so 2 equals 0."},{"Start":"05:08.110 ","End":"05:11.075","Text":"That\u0027s certainly not possible."},{"Start":"05:11.075 ","End":"05:16.280","Text":"Basically, there is no way of getting this into this form."},{"Start":"05:16.280 ","End":"05:18.500","Text":"If it doesn\u0027t work like this,"},{"Start":"05:18.500 ","End":"05:21.260","Text":"and hence it won\u0027t work with the A and the B,"},{"Start":"05:21.260 ","End":"05:28.685","Text":"what is the basic form for case 2 when we have something squared in the bottom?"},{"Start":"05:28.685 ","End":"05:34.730","Text":"The answer is not very far from what we did like here."},{"Start":"05:34.730 ","End":"05:36.740","Text":"We do have an A and a B,"},{"Start":"05:36.740 ","End":"05:44.660","Text":"but what we try is we try for x plus 1 over x minus 1 squared to B."},{"Start":"05:44.660 ","End":"05:46.355","Text":"We start out all right,"},{"Start":"05:46.355 ","End":"05:49.234","Text":"A over x minus 1."},{"Start":"05:49.234 ","End":"05:53.150","Text":"But we have a B not over x minus 1,"},{"Start":"05:53.150 ","End":"05:56.045","Text":"but over x minus 1 squared."},{"Start":"05:56.045 ","End":"05:59.800","Text":"This basic form is what does the trick each time."},{"Start":"05:59.800 ","End":"06:01.685","Text":"We\u0027ll generalize it in a moment."},{"Start":"06:01.685 ","End":"06:02.720","Text":"But in our case,"},{"Start":"06:02.720 ","End":"06:04.400","Text":"this is what we\u0027re looking for,"},{"Start":"06:04.400 ","End":"06:08.510","Text":"partial fractions and we\u0027re going to use the same technique as before."},{"Start":"06:08.510 ","End":"06:10.070","Text":"We\u0027re going to get a common denominator,"},{"Start":"06:10.070 ","End":"06:11.810","Text":"get rid of the denominator,"},{"Start":"06:11.810 ","End":"06:13.860","Text":"and then substitute a couple of values,"},{"Start":"06:13.860 ","End":"06:15.830","Text":"we\u0027ll get 2 equations and 2 unknowns,"},{"Start":"06:15.830 ","End":"06:19.050","Text":"A and B. Let\u0027s start that."},{"Start":"06:19.050 ","End":"06:22.580","Text":"I\u0027m going to put this over a common denominator,"},{"Start":"06:22.580 ","End":"06:25.550","Text":"I\u0027m going to put it all over x minus 1 squared."},{"Start":"06:25.550 ","End":"06:29.420","Text":"The only thing I have to do is with this term here,"},{"Start":"06:29.420 ","End":"06:36.710","Text":"I have to multiply numerator and denominator by an extra x minus 1 here and here."},{"Start":"06:36.710 ","End":"06:42.630","Text":"What I\u0027m going to get is x plus 1 is"},{"Start":"06:42.630 ","End":"06:48.220","Text":"equal to A times x minus 1 plus B."},{"Start":"06:48.220 ","End":"06:51.530","Text":"Basically, I\u0027ve just thrown out the common denominator,"},{"Start":"06:51.530 ","End":"06:52.940","Text":"x minus 1 squared,"},{"Start":"06:52.940 ","End":"06:54.665","Text":"and this is what I\u0027m left with."},{"Start":"06:54.665 ","End":"07:00.090","Text":"Now I can put in whatever I want for x,"},{"Start":"07:00.090 ","End":"07:02.895","Text":"and I\u0027ll put 2 different values."},{"Start":"07:02.895 ","End":"07:08.090","Text":"Let\u0027s try first of all, the easiest I guess is x equals 0."},{"Start":"07:08.090 ","End":"07:09.710","Text":"If x is 0,"},{"Start":"07:09.710 ","End":"07:10.885","Text":"let\u0027s see what we get."},{"Start":"07:10.885 ","End":"07:12.440","Text":"I\u0027ll put in x equals 0,"},{"Start":"07:12.440 ","End":"07:13.535","Text":"then we\u0027ll put something else."},{"Start":"07:13.535 ","End":"07:18.020","Text":"If 0, I get 1 equals A times minus 1,"},{"Start":"07:18.020 ","End":"07:21.760","Text":"which is minus A plus B."},{"Start":"07:21.760 ","End":"07:24.139","Text":"If I put x equals 1,"},{"Start":"07:24.139 ","End":"07:27.290","Text":"that might be good because I\u0027ll get rid of this term."},{"Start":"07:27.290 ","End":"07:29.945","Text":"I\u0027ll put x equals 1,"},{"Start":"07:29.945 ","End":"07:33.245","Text":"and then I get 1 plus 1 is"},{"Start":"07:33.245 ","End":"07:41.040","Text":"2 equals B. I\u0027ve got B equals 2,"},{"Start":"07:41.040 ","End":"07:51.765","Text":"so I can just plug that in here and get from here 1 equals minus A plus 2."},{"Start":"07:51.765 ","End":"07:55.980","Text":"That gives me that A equals 1."},{"Start":"07:55.980 ","End":"08:00.060","Text":"I\u0027ve got A, if I put A over here, it\u0027s 2 minus 1."},{"Start":"08:00.060 ","End":"08:04.095","Text":"A is 1, B equals 2."},{"Start":"08:04.095 ","End":"08:06.990","Text":"That\u0027s the solution for this."},{"Start":"08:06.990 ","End":"08:16.325","Text":"What it gives us is that the partial fraction decomposition of this is 1 over x minus 1,"},{"Start":"08:16.325 ","End":"08:27.120","Text":"because A is 1, plus 2 over x minus 1 squared."},{"Start":"08:27.850 ","End":"08:31.675","Text":"This is the partial decomposition."},{"Start":"08:31.675 ","End":"08:36.110","Text":"This integral, after all this processing comes out to"},{"Start":"08:36.110 ","End":"08:41.665","Text":"be the integral of 1 over x minus 1,"},{"Start":"08:41.665 ","End":"08:50.570","Text":"dx plus the integral,"},{"Start":"08:50.570 ","End":"08:53.255","Text":"and I can take the 2 outside the brackets,"},{"Start":"08:53.255 ","End":"09:01.625","Text":"1 over x minus 1 squared dx."},{"Start":"09:01.625 ","End":"09:06.545","Text":"Now, I want to do a couple of computations at the side and we\u0027ll need them a lot."},{"Start":"09:06.545 ","End":"09:08.690","Text":"This is just a reminder really,"},{"Start":"09:08.690 ","End":"09:10.790","Text":"that the integral of"},{"Start":"09:10.790 ","End":"09:16.520","Text":"1 over x dx"},{"Start":"09:16.520 ","End":"09:21.740","Text":"is equal to the natural log of absolute value of x plus a constant,"},{"Start":"09:21.740 ","End":"09:23.060","Text":"which I haven\u0027t written."},{"Start":"09:23.060 ","End":"09:32.655","Text":"The integral of 1 over x squared dx is minus 1 over x,"},{"Start":"09:32.655 ","End":"09:35.110","Text":"also plus a constant."},{"Start":"09:35.110 ","End":"09:38.240","Text":"Because if I differentiate 1 over x,"},{"Start":"09:38.240 ","End":"09:39.920","Text":"I get minus 1 over x squared."},{"Start":"09:39.920 ","End":"09:42.165","Text":"That\u0027s an easy differentiation,"},{"Start":"09:42.165 ","End":"09:43.890","Text":"and so I need a minus."},{"Start":"09:43.890 ","End":"09:45.140","Text":"Or you can look at it the other way."},{"Start":"09:45.140 ","End":"09:48.170","Text":"This is x^minus 2,"},{"Start":"09:48.170 ","End":"09:49.400","Text":"and when I integrate it,"},{"Start":"09:49.400 ","End":"09:52.370","Text":"I get x^minus 1 over minus 1,"},{"Start":"09:52.370 ","End":"09:54.395","Text":"which is equal to this."},{"Start":"09:54.395 ","End":"09:58.430","Text":"These 2 will help us here because the same thing works if I"},{"Start":"09:58.430 ","End":"10:02.090","Text":"put instead of x as x minus A."},{"Start":"10:02.090 ","End":"10:05.750","Text":"Basically what I\u0027m saying is the important thing from these is"},{"Start":"10:05.750 ","End":"10:10.040","Text":"that integral of 1 over x minus A"},{"Start":"10:10.040 ","End":"10:14.210","Text":"squared dx is very similar to 1 over x"},{"Start":"10:14.210 ","End":"10:19.010","Text":"squared because the derivative of x minus A is also 1 in a derivative."},{"Start":"10:19.010 ","End":"10:29.320","Text":"This will just equal minus 1 over x minus A plus a constant."},{"Start":"10:29.860 ","End":"10:32.900","Text":"Now we get back here."},{"Start":"10:32.900 ","End":"10:38.090","Text":"What we get is the integral of this will be"},{"Start":"10:38.090 ","End":"10:47.115","Text":"natural log of x minus 1 in absolute values, plus twice."},{"Start":"10:47.115 ","End":"10:49.990","Text":"From here I get that it\u0027s minus"},{"Start":"10:49.990 ","End":"10:58.050","Text":"1 over x minus"},{"Start":"10:58.050 ","End":"11:01.990","Text":"1 plus a constant."},{"Start":"11:02.710 ","End":"11:05.285","Text":"This is the answer just,"},{"Start":"11:05.285 ","End":"11:10.415","Text":"I can throw the 2 into the numerator here and put the minus outside."},{"Start":"11:10.415 ","End":"11:14.925","Text":"Let me just do that there."},{"Start":"11:14.925 ","End":"11:19.485","Text":"I just put the 2 here and put the minus here. That\u0027s the answer."},{"Start":"11:19.485 ","End":"11:22.730","Text":"I think in general, this is worth remembering because it will occur"},{"Start":"11:22.730 ","End":"11:27.100","Text":"in all this kinds of exercise."},{"Start":"11:28.490 ","End":"11:36.435","Text":"Yeah, the integral of 1 over x minus a squared dx is just minus 1 over x minus a."},{"Start":"11:36.435 ","End":"11:39.970","Text":"Next, I\u0027m going to summarize the process."},{"Start":"11:40.280 ","End":"11:44.850","Text":"Here\u0027s a summary or recipe if you like,"},{"Start":"11:44.850 ","End":"11:47.475","Text":"for the Case 2."},{"Start":"11:47.475 ","End":"11:53.940","Text":"We start with the integral in the form mx plus n"},{"Start":"11:53.940 ","End":"12:02.520","Text":"over x squared plus bx plus c. Now,"},{"Start":"12:02.520 ","End":"12:10.500","Text":"Case 2 means that this denominator has a single root,"},{"Start":"12:10.500 ","End":"12:13.710","Text":"and that root is x equals, let\u0027s say k,"},{"Start":"12:13.710 ","End":"12:19.575","Text":"some number, and there\u0027s a dx here, I sometimes forget."},{"Start":"12:19.575 ","End":"12:22.290","Text":"Then we bring this into the form."},{"Start":"12:22.290 ","End":"12:27.135","Text":"The same numerator, mx plus n, but the denominator,"},{"Start":"12:27.135 ","End":"12:29.850","Text":"because it\u0027s a single root, x equals k,"},{"Start":"12:29.850 ","End":"12:36.190","Text":"comes out to be x minus k squared dx."},{"Start":"12:36.380 ","End":"12:41.475","Text":"Then when we do the partial fraction on the function part,"},{"Start":"12:41.475 ","End":"12:48.510","Text":"the rational function, what we get is similar yet different to the Case 1."},{"Start":"12:48.510 ","End":"12:58.780","Text":"In Case 1, we had A over x minus 1 of the roots plus B over x minus the other root."},{"Start":"12:58.780 ","End":"13:01.470","Text":"But here there is no other root that both the same,"},{"Start":"13:01.470 ","End":"13:03.540","Text":"so we have to remember that in this case,"},{"Start":"13:03.540 ","End":"13:05.010","Text":"we put a squared here,"},{"Start":"13:05.010 ","End":"13:08.790","Text":"this is the basic form of the partial fraction."},{"Start":"13:08.790 ","End":"13:13.695","Text":"Then it put the brackets here and dx."},{"Start":"13:13.695 ","End":"13:19.845","Text":"This thing comes out to be familiar with the A over x minus k."},{"Start":"13:19.845 ","End":"13:26.310","Text":"That\u0027s the natural logarithm of x minus k or absolute value."},{"Start":"13:26.310 ","End":"13:28.110","Text":"But the second part,"},{"Start":"13:28.110 ","End":"13:29.805","Text":"like I just showed you,"},{"Start":"13:29.805 ","End":"13:36.675","Text":"comes out to be minus B over x minus k,"},{"Start":"13:36.675 ","End":"13:40.035","Text":"and finally plus a constant of integration."},{"Start":"13:40.035 ","End":"13:43.949","Text":"This is the basic solution,"},{"Start":"13:43.949 ","End":"13:47.490","Text":"I forgot a constant A here."},{"Start":"13:47.490 ","End":"13:51.480","Text":"It\u0027s my own fault, I should have put in an extra step here."},{"Start":"13:51.480 ","End":"13:55.560","Text":"I should have written this as the sum of the integrals,"},{"Start":"13:55.560 ","End":"13:57.060","Text":"but take the constant out,"},{"Start":"13:57.060 ","End":"14:06.480","Text":"so I\u0027d get A times the integral of 1 over x minus k dx plus"},{"Start":"14:06.480 ","End":"14:16.770","Text":"B times the integral of 1 over x minus k squared dx."},{"Start":"14:16.770 ","End":"14:19.920","Text":"Then we would say 1 over x minus k,"},{"Start":"14:19.920 ","End":"14:24.855","Text":"its integral is natural log absolute x minus k. This integral,"},{"Start":"14:24.855 ","End":"14:26.655","Text":"like we showed before,"},{"Start":"14:26.655 ","End":"14:29.940","Text":"is minus 1 over x minus k,"},{"Start":"14:29.940 ","End":"14:33.585","Text":"and B goes in combines with the 1,"},{"Start":"14:33.585 ","End":"14:37.725","Text":"and so this is the answer that we get."},{"Start":"14:37.725 ","End":"14:42.270","Text":"Continuing with the integration of rational functions,"},{"Start":"14:42.270 ","End":"14:49.120","Text":"basic Case 2 is our last example and then we finished with this clip."},{"Start":"14:49.640 ","End":"14:54.090","Text":"The most important thing is the denominator and how many roots it has."},{"Start":"14:54.090 ","End":"14:59.730","Text":"Basically what we have to do is to factorize the denominator."},{"Start":"14:59.730 ","End":"15:04.090","Text":"Factorize or let\u0027s write it as the denom."},{"Start":"15:04.910 ","End":"15:07.335","Text":"If you have a show where by,"},{"Start":"15:07.335 ","End":"15:11.820","Text":"you could probably see straight away that it\u0027s as x plus 2 all squared."},{"Start":"15:11.820 ","End":"15:19.395","Text":"Or if not, you could take the equation x squared plus 4x plus 4 equals 0 and solve it."},{"Start":"15:19.395 ","End":"15:20.790","Text":"I\u0027m not going to solve it for you,"},{"Start":"15:20.790 ","End":"15:23.550","Text":"but you get that there is only 1 solution,"},{"Start":"15:23.550 ","End":"15:28.020","Text":"and that is that x equals minus 2."},{"Start":"15:28.020 ","End":"15:35.760","Text":"In general, when we have x squared plus bx plus c equals 0 is an equation,"},{"Start":"15:35.760 ","End":"15:38.910","Text":"and if it only has 1 solution,"},{"Start":"15:38.910 ","End":"15:42.345","Text":"x is equal to k,"},{"Start":"15:42.345 ","End":"15:50.835","Text":"then this thing factorizes into x minus k squared."},{"Start":"15:50.835 ","End":"15:58.590","Text":"In our case, we get x minus minus 2 squared,"},{"Start":"15:58.590 ","End":"16:06.105","Text":"and so we can factorize the denominator as x plus 2 squared,"},{"Start":"16:06.105 ","End":"16:11.145","Text":"and write that now instead of the original."},{"Start":"16:11.145 ","End":"16:17.460","Text":"The next step is to decompose this into partial fractions."},{"Start":"16:17.460 ","End":"16:21.105","Text":"I\u0027ll just call the next heading partial fractions."},{"Start":"16:21.105 ","End":"16:24.270","Text":"Partial fractions has nothing to do with integration,"},{"Start":"16:24.270 ","End":"16:26.565","Text":"it\u0027s just an algebraic thing."},{"Start":"16:26.565 ","End":"16:36.515","Text":"We want to write 4x plus 18 over x plus 2 squared as equal to,"},{"Start":"16:36.515 ","End":"16:43.845","Text":"and we discovered that the correct form is some constant A over x plus"},{"Start":"16:43.845 ","End":"16:52.320","Text":"2 plus another constant B over x plus 2 squared."},{"Start":"16:52.320 ","End":"16:57.270","Text":"This is like a reverse process because"},{"Start":"16:57.270 ","End":"16:59.520","Text":"normally you would start with this and this and you\u0027d"},{"Start":"16:59.520 ","End":"17:02.250","Text":"put them under a common denominator and get this."},{"Start":"17:02.250 ","End":"17:03.795","Text":"But we\u0027re doing it backwards,"},{"Start":"17:03.795 ","End":"17:06.480","Text":"and the way to solve this partial fraction"},{"Start":"17:06.480 ","End":"17:09.330","Text":"is to put everything over a common denominator,"},{"Start":"17:09.330 ","End":"17:11.220","Text":"x plus 2 squared."},{"Start":"17:11.220 ","End":"17:14.895","Text":"Now this already is the common denominator and so is this."},{"Start":"17:14.895 ","End":"17:17.325","Text":"All that\u0027s missing is the squared here,"},{"Start":"17:17.325 ","End":"17:21.480","Text":"so we multiply top and bottom by x plus 2, and then we get,"},{"Start":"17:21.480 ","End":"17:26.700","Text":"if we just take the numerators that 4x plus 18 is"},{"Start":"17:26.700 ","End":"17:33.400","Text":"equal to A times the missing x plus 2 here, plus B."},{"Start":"17:33.470 ","End":"17:37.049","Text":"This is not just an equality,"},{"Start":"17:37.049 ","End":"17:41.174","Text":"it\u0027s an identity which means that it\u0027s true for all x,"},{"Start":"17:41.174 ","End":"17:44.624","Text":"I can substitute whatever value of x I like,"},{"Start":"17:44.624 ","End":"17:54.150","Text":"and so for starters I\u0027ll put in x equals minus 2 because that will make this term 0."},{"Start":"17:54.150 ","End":"17:55.935","Text":"If x is minus 2,"},{"Start":"17:55.935 ","End":"18:01.710","Text":"I get minus 2 times 4 is minus 8 plus 18 is 10."},{"Start":"18:01.710 ","End":"18:08.655","Text":"This equals 0 times A, or A times 0,"},{"Start":"18:08.655 ","End":"18:16.500","Text":"plus B, so that gives me that B is equal to 10."},{"Start":"18:16.500 ","End":"18:21.645","Text":"Now if I substitute, for example,"},{"Start":"18:21.645 ","End":"18:29.295","Text":"x equals 0, then we get if x is 0 we just get 18 here."},{"Start":"18:29.295 ","End":"18:37.260","Text":"0 plus 18 and 0 here means 0 plus 2 is 2 times A is 2A,"},{"Start":"18:37.260 ","End":"18:41.070","Text":"plus B, but B we already know is 10,"},{"Start":"18:41.070 ","End":"18:42.705","Text":"so I can do that."},{"Start":"18:42.705 ","End":"18:45.600","Text":"Then if I extract A,"},{"Start":"18:45.600 ","End":"18:51.705","Text":"I get that A is equal to 18 minus 10 is 8 divided by 2 is 4,"},{"Start":"18:51.705 ","End":"18:53.310","Text":"so we have B is 10,"},{"Start":"18:53.310 ","End":"18:57.930","Text":"A is 4, and now I can continue with the integration."},{"Start":"18:57.930 ","End":"19:01.395","Text":"I\u0027ll just copy this thing over here,"},{"Start":"19:01.395 ","End":"19:10.319","Text":"so the integral of 4x plus 18 over x plus 2 squared dx equals."},{"Start":"19:10.319 ","End":"19:14.205","Text":"Now use the partial fraction decomposition instead of this,"},{"Start":"19:14.205 ","End":"19:18.225","Text":"so what we have is A over x plus 2,"},{"Start":"19:18.225 ","End":"19:22.170","Text":"which is 4 over x plus"},{"Start":"19:22.170 ","End":"19:31.050","Text":"2 plus B is 10 over x plus 2 squared,"},{"Start":"19:31.050 ","End":"19:34.230","Text":"and all this dx."},{"Start":"19:34.230 ","End":"19:37.830","Text":"Now this is equal to I just"},{"Start":"19:37.830 ","End":"19:41.250","Text":"take the sum of the integrals and the constants income outside,"},{"Start":"19:41.250 ","End":"19:50.400","Text":"so it\u0027s 4 times the integral of 1 over x plus 2 dx plus 10"},{"Start":"19:50.400 ","End":"20:00.240","Text":"times the integral of 1 over x plus 2 squared dx,"},{"Start":"20:00.240 ","End":"20:06.045","Text":"and this we know what it equals because we practice before."},{"Start":"20:06.045 ","End":"20:07.920","Text":"This 1 too write again,"},{"Start":"20:07.920 ","End":"20:12.510","Text":"the formula that I mentioned earlier is that in general,"},{"Start":"20:12.510 ","End":"20:19.440","Text":"the integral of 1 over x minus"},{"Start":"20:19.440 ","End":"20:23.760","Text":"a squared dx is equal"},{"Start":"20:23.760 ","End":"20:30.540","Text":"to minus 1 over x minus a."},{"Start":"20:30.540 ","End":"20:36.015","Text":"Well, if we wanted, we can put a plus c, not important here."},{"Start":"20:36.015 ","End":"20:40.640","Text":"So we get 4 times for this bit,"},{"Start":"20:40.640 ","End":"20:45.170","Text":"we get the natural logarithm of x plus"},{"Start":"20:45.170 ","End":"20:52.680","Text":"2 plus 10 times the integral of this bit is what we said here,"},{"Start":"20:52.680 ","End":"20:56.470","Text":"minus 1 over x plus 2,"},{"Start":"20:56.960 ","End":"21:00.420","Text":"and plus the constant,"},{"Start":"21:00.420 ","End":"21:02.265","Text":"which we\u0027ll do at the very end."},{"Start":"21:02.265 ","End":"21:07.200","Text":"Which is equal to 4 natural log of x plus"},{"Start":"21:07.200 ","End":"21:16.905","Text":"2 minus 10 over x plus 2 plus the constant."},{"Start":"21:16.905 ","End":"21:20.140","Text":"That\u0027s the end of it."},{"Start":"21:21.200 ","End":"21:23.805","Text":"That\u0027s the solution."},{"Start":"21:23.805 ","End":"21:26.115","Text":"I\u0027m done with Case 2."},{"Start":"21:26.115 ","End":"21:29.110","Text":"Next clip, we\u0027ll do Case 3."}],"ID":4492},{"Watched":false,"Name":"Basic Case III","Duration":"29m 22s","ChapterTopicVideoID":4484,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.330","Text":"Continuing with the integration of rational functions, the basic case."},{"Start":"00:06.330 ","End":"00:08.340","Text":"We\u0027ll take Case 3 ,"},{"Start":"00:08.340 ","End":"00:10.725","Text":"which is the last of the 3."},{"Start":"00:10.725 ","End":"00:17.415","Text":"I\u0027ll just remind you that what we\u0027re dealing with is the integral of"},{"Start":"00:17.415 ","End":"00:22.320","Text":"a linear polynomial like mx plus n over"},{"Start":"00:22.320 ","End":"00:28.540","Text":"a quadratic polynomial with coefficient 1 of x squared."},{"Start":"00:29.230 ","End":"00:32.840","Text":"This is the general Basic Case."},{"Start":"00:32.840 ","End":"00:37.790","Text":"Case 3 is when the denominator has no roots."},{"Start":"00:37.790 ","End":"00:43.435","Text":"This thing, that denominator has no roots."},{"Start":"00:43.435 ","End":"00:48.770","Text":"Means the equation that this equals 0 has no solutions in other words."},{"Start":"00:48.770 ","End":"00:51.785","Text":"I\u0027ll start right away with an example."},{"Start":"00:51.785 ","End":"00:58.020","Text":"Let\u0027s take as an example the integral of 1 over x"},{"Start":"00:58.020 ","End":"01:06.410","Text":"squared plus x plus 4 dx."},{"Start":"01:06.410 ","End":"01:09.290","Text":"They might say this doesn\u0027t quite look like this,"},{"Start":"01:09.290 ","End":"01:12.830","Text":"but if m is 0 and n is 1,"},{"Start":"01:12.830 ","End":"01:15.180","Text":"then this is what you\u0027ll get."},{"Start":"01:16.180 ","End":"01:21.155","Text":"You can check that x squared plus x plus 4 has no roots."},{"Start":"01:21.155 ","End":"01:25.160","Text":"We can easily check that it has no solutions by checking the discriminant,"},{"Start":"01:25.160 ","End":"01:29.560","Text":"which is b squared minus 4ac."},{"Start":"01:29.560 ","End":"01:32.910","Text":"In this case, b is 1,"},{"Start":"01:32.910 ","End":"01:35.475","Text":"a is 1, and c is 4,"},{"Start":"01:35.475 ","End":"01:42.605","Text":"so we get 1 squared minus 4 times 1 times 4."},{"Start":"01:42.605 ","End":"01:45.395","Text":"That clearly comes out negative."},{"Start":"01:45.395 ","End":"01:47.225","Text":"When the discriminant is negative,"},{"Start":"01:47.225 ","End":"01:48.860","Text":"there are no solutions."},{"Start":"01:48.860 ","End":"01:51.840","Text":"Because the denominator has no roots,"},{"Start":"01:51.840 ","End":"01:53.490","Text":"we can\u0027t factorize it."},{"Start":"01:53.490 ","End":"01:55.310","Text":"If we can\u0027t factorize it,"},{"Start":"01:55.310 ","End":"02:00.050","Text":"then the method of decomposition into partial fractions is of no use, it won\u0027t work."},{"Start":"02:00.050 ","End":"02:02.615","Text":"We need something else,"},{"Start":"02:02.615 ","End":"02:04.520","Text":"a different set of tools."},{"Start":"02:04.520 ","End":"02:07.075","Text":"What we\u0027re going to do is this."},{"Start":"02:07.075 ","End":"02:11.115","Text":"We\u0027re going to sub-divide Case 3 into 2."},{"Start":"02:11.115 ","End":"02:15.760","Text":"It\u0027s going to be a Case 3 short and a Case 3 long."},{"Start":"02:16.040 ","End":"02:26.339","Text":"We take Case 3 and we divide it into short and long."},{"Start":"02:26.440 ","End":"02:33.420","Text":"Short and the long is just the regular case."},{"Start":"02:33.830 ","End":"02:39.110","Text":"But the short is a special case when b is 0,"},{"Start":"02:39.110 ","End":"02:40.775","Text":"there is no x term."},{"Start":"02:40.775 ","End":"02:46.625","Text":"We get the integral of mx plus"},{"Start":"02:46.625 ","End":"02:55.200","Text":"n over x squared plus c. This is called the short case, and you\u0027ll see why."},{"Start":"02:55.200 ","End":"02:58.095","Text":"Basically because it\u0027s shorter to solve than this."},{"Start":"02:58.095 ","End":"03:06.530","Text":"Of course, c has to be positive because if it\u0027s not positive,"},{"Start":"03:06.530 ","End":"03:08.525","Text":"then we can factorize it."},{"Start":"03:08.525 ","End":"03:11.555","Text":"If it\u0027s negative x squared minus something,"},{"Start":"03:11.555 ","End":"03:13.700","Text":"we can decompose it into x minus"},{"Start":"03:13.700 ","End":"03:16.340","Text":"the square root of that something and x plus the square root."},{"Start":"03:16.340 ","End":"03:22.000","Text":"Brief c is bigger than 0 and there\u0027s no x term."},{"Start":"03:22.000 ","End":"03:24.810","Text":"I will start with the short case."},{"Start":"03:24.810 ","End":"03:31.640","Text":"This happens not to be of the short case because it does have an x here."},{"Start":"03:31.640 ","End":"03:35.755","Text":"I\u0027m going to give some examples of short and long."},{"Start":"03:35.755 ","End":"03:38.175","Text":"They\u0027re all in Case III,"},{"Start":"03:38.175 ","End":"03:41.185","Text":"examples in Case III,"},{"Start":"03:41.185 ","End":"03:45.710","Text":"which we\u0027ve now subdivided into short."},{"Start":"03:45.710 ","End":"03:54.060","Text":"An example of this will be 1 over x squared plus 1."},{"Start":"03:54.060 ","End":"03:57.920","Text":"Well, I\u0027ll write the whole thing, the integral of that, dx."},{"Start":"03:57.920 ","End":"04:02.795","Text":"Another example will be integral of"},{"Start":"04:02.795 ","End":"04:12.995","Text":"4x minus 10 over x squared plus 100."},{"Start":"04:12.995 ","End":"04:16.055","Text":"Doesn\u0027t matter about the numerator."},{"Start":"04:16.055 ","End":"04:20.885","Text":"What\u0027s important is that the denominator is x squared plus something,"},{"Start":"04:20.885 ","End":"04:23.390","Text":"that makes it the short case."},{"Start":"04:23.390 ","End":"04:27.590","Text":"Long is what we\u0027ve been doing all along, so to speak."},{"Start":"04:27.590 ","End":"04:30.830","Text":"This one, we can just repeat that one."},{"Start":"04:30.830 ","End":"04:33.520","Text":"We know that one\u0027s a long case,"},{"Start":"04:33.520 ","End":"04:36.440","Text":"x squared plus x plus 4,"},{"Start":"04:36.440 ","End":"04:41.810","Text":"because the denominator doesn\u0027t have any roots and the x term is present."},{"Start":"04:41.810 ","End":"04:47.730","Text":"Another example 4x minus 10"},{"Start":"04:47.730 ","End":"04:58.260","Text":"over x squared plus 2x plus 11, and many others."},{"Start":"04:58.260 ","End":"05:02.535","Text":"First of all, we\u0027ll try one of the short cases."},{"Start":"05:02.535 ","End":"05:06.030","Text":"Just occurred to me, this is a bad title."},{"Start":"05:06.030 ","End":"05:08.220","Text":"Because we\u0027re going to do the examples in a moment,"},{"Start":"05:08.220 ","End":"05:10.250","Text":"this should be called definition,"},{"Start":"05:10.250 ","End":"05:13.160","Text":"I guess. That\u0027s better."},{"Start":"05:13.160 ","End":"05:17.335","Text":"Now we are going to do some examples of the short case."},{"Start":"05:17.335 ","End":"05:22.130","Text":"Before the example, I want to give a formula that we\u0027re going to use a lot."},{"Start":"05:22.130 ","End":"05:24.425","Text":"It\u0027s familiar, should be familiar to you."},{"Start":"05:24.425 ","End":"05:26.330","Text":"It\u0027s that the integral,"},{"Start":"05:26.330 ","End":"05:31.235","Text":"if we have the derivative of a function over the function itself,"},{"Start":"05:31.235 ","End":"05:40.870","Text":"dx, then this is equal to the natural log of absolute value of f plus a constant."},{"Start":"05:40.870 ","End":"05:45.020","Text":"Now, here\u0027s an example that we can solve already."},{"Start":"05:45.020 ","End":"05:50.449","Text":"It\u0027s certainly Case III and the short case because there\u0027s no x here,"},{"Start":"05:50.449 ","End":"05:53.760","Text":"and it has no roots because it\u0027s a plus here."},{"Start":"05:55.430 ","End":"05:57.990","Text":"Now look at this formula,"},{"Start":"05:57.990 ","End":"06:00.860","Text":"2x is the derivative of x squared plus 4,"},{"Start":"06:00.860 ","End":"06:04.070","Text":"so we can immediately answer by saying that this is"},{"Start":"06:04.070 ","End":"06:10.460","Text":"natural log of x squared plus 4 plus constant."},{"Start":"06:10.460 ","End":"06:15.845","Text":"Next example, the integral"},{"Start":"06:15.845 ","End":"06:22.790","Text":"of x over x squared plus 4."},{"Start":"06:22.790 ","End":"06:28.050","Text":"Sorry, let\u0027s make it x squared plus 1,"},{"Start":"06:28.050 ","End":"06:30.100","Text":"just for a change."},{"Start":"06:30.470 ","End":"06:34.220","Text":"The derivative of the denominator is not the numerator."},{"Start":"06:34.220 ","End":"06:36.155","Text":"We want 2x, and here we have x."},{"Start":"06:36.155 ","End":"06:37.910","Text":"What are we going to do?"},{"Start":"06:37.910 ","End":"06:39.650","Text":"Well, that\u0027s pretty clear."},{"Start":"06:39.650 ","End":"06:44.825","Text":"We\u0027ll put a 2 here and we\u0027ll compensate by putting a 1/2 in front."},{"Start":"06:44.825 ","End":"06:53.130","Text":"This is 1/2 the integral of 2x over x squared plus 1 dx."},{"Start":"06:53.130 ","End":"06:54.395","Text":"I put the 2 here,"},{"Start":"06:54.395 ","End":"06:59.240","Text":"but I compensated by dividing by 2 and I put the 1/2 outside the integral sign."},{"Start":"06:59.240 ","End":"07:02.825","Text":"Now, this is equal to the 1/2 was here,"},{"Start":"07:02.825 ","End":"07:04.955","Text":"and then we just have the formula,"},{"Start":"07:04.955 ","End":"07:15.025","Text":"natural log of absolute value of x squared plus 1 and again plus C. Now,"},{"Start":"07:15.025 ","End":"07:22.265","Text":"how about I take the integral of 7 x over x squared plus 1?"},{"Start":"07:22.265 ","End":"07:24.310","Text":"Let\u0027s keep it the same as above."},{"Start":"07:24.310 ","End":"07:26.220","Text":"What will this equal?"},{"Start":"07:26.220 ","End":"07:27.945","Text":"Now, we wanted 2 here."},{"Start":"07:27.945 ","End":"07:29.480","Text":"Well, it\u0027s fairly clear."},{"Start":"07:29.480 ","End":"07:33.460","Text":"We take the 7 outside of the integral sign."},{"Start":"07:33.460 ","End":"07:35.380","Text":"Now we use the same trick as here,"},{"Start":"07:35.380 ","End":"07:38.000","Text":"is dividing by 2 and multiplying by 2,"},{"Start":"07:38.000 ","End":"07:45.175","Text":"so we get 7 over 2 times 2x over x squared plus 1."},{"Start":"07:45.175 ","End":"07:50.000","Text":"This time we get the same thing except that it\u0027s 7 over 2 natural log"},{"Start":"07:50.000 ","End":"07:54.235","Text":"of absolute value of x squared plus 1 plus a constant."},{"Start":"07:54.235 ","End":"07:58.550","Text":"Now, I see that I\u0027m going to be needing to use the letter C for what I\u0027m about to write."},{"Start":"07:58.550 ","End":"08:03.425","Text":"I\u0027ll tell you what, I\u0027m going to change this letter C to K. I\u0027m sure you don\u0027t mind."},{"Start":"08:03.425 ","End":"08:08.434","Text":"The reason I need the C is for the following generalization of this."},{"Start":"08:08.434 ","End":"08:14.245","Text":"The integral of mx"},{"Start":"08:14.245 ","End":"08:21.540","Text":"over x squared plus c dx is equal to,"},{"Start":"08:21.540 ","End":"08:25.095","Text":"now think of the m as 7 and the c as the 1,"},{"Start":"08:25.095 ","End":"08:28.910","Text":"so I get m"},{"Start":"08:28.910 ","End":"08:37.070","Text":"over 2 times natural log of x squared plus c plus k,"},{"Start":"08:37.070 ","End":"08:41.610","Text":"the constant of integration because I\u0027ve used up c already here."},{"Start":"08:42.010 ","End":"08:48.350","Text":"This means that we basically solved short Case 3 whenever n is 0."},{"Start":"08:48.350 ","End":"08:49.580","Text":"We just have the mx,"},{"Start":"08:49.580 ","End":"08:54.100","Text":"but we don\u0027t have the n. We\u0027re still on short Case 3,"},{"Start":"08:54.100 ","End":"08:56.090","Text":"but we still haven\u0027t got to the general case."},{"Start":"08:56.090 ","End":"09:02.525","Text":"We have mx, but we want to lead up to mx plus n. First of all,"},{"Start":"09:02.525 ","End":"09:05.970","Text":"I\u0027ll show you how to do it when there\u0027s just a constant here"},{"Start":"09:05.970 ","End":"09:09.410","Text":"and then we\u0027ll combine the 2 cases and get"},{"Start":"09:09.410 ","End":"09:13.310","Text":"the full mx plus n. I want to write down"},{"Start":"09:13.310 ","End":"09:18.300","Text":"another formula which is going to really help us here."},{"Start":"09:18.670 ","End":"09:27.425","Text":"For example, if I have the integral of 1 over x squared plus 1,"},{"Start":"09:27.425 ","End":"09:29.640","Text":"then c is equal to 1,"},{"Start":"09:29.640 ","End":"09:33.675","Text":"so we get 1 over the square root of 1 is 1."},{"Start":"09:33.675 ","End":"09:38.370","Text":"We just get the arctangent of"},{"Start":"09:38.370 ","End":"09:48.295","Text":"x plus k. If we were to take 1 over x squared plus 4,"},{"Start":"09:48.295 ","End":"09:51.420","Text":"then c is 4,"},{"Start":"09:51.420 ","End":"09:54.055","Text":"square root of 4 is 2."},{"Start":"09:54.055 ","End":"09:56.180","Text":"By the way, c is always positive,"},{"Start":"09:56.180 ","End":"09:57.605","Text":"like I mentioned here,"},{"Start":"09:57.605 ","End":"09:59.090","Text":"because if she wasn\u0027t positive,"},{"Start":"09:59.090 ","End":"10:00.350","Text":"we could factorize this."},{"Start":"10:00.350 ","End":"10:02.420","Text":"So it does have a square root."},{"Start":"10:02.420 ","End":"10:09.260","Text":"Anyway, this is equal to 1/2 arctangent of"},{"Start":"10:09.260 ","End":"10:16.530","Text":"x over 2 plus k. So far we\u0027ve taken 1 on the numerator,"},{"Start":"10:16.530 ","End":"10:19.490","Text":"how about let\u0027s take 4 on the numerator."},{"Start":"10:19.490 ","End":"10:25.080","Text":"So 4 over x squared plus, let\u0027s say 10."},{"Start":"10:25.650 ","End":"10:29.620","Text":"Then square root of 10 is just square root of 10."},{"Start":"10:29.620 ","End":"10:33.025","Text":"I write 1 over the square root of 10."},{"Start":"10:33.025 ","End":"10:38.725","Text":"But I also take the 4 outside the integral so I can put the 4 here."},{"Start":"10:38.725 ","End":"10:41.590","Text":"4 over square root of 10,"},{"Start":"10:41.590 ","End":"10:48.040","Text":"arc-tangent of x over square root of 10."},{"Start":"10:48.040 ","End":"10:52.944","Text":"I think we\u0027re now ready to go on to the general case."},{"Start":"10:52.944 ","End":"10:57.640","Text":"Let me just say that, just like I generalized here,"},{"Start":"10:57.640 ","End":"11:05.560","Text":"I could actually generalize the case where m is 0 but n is not."},{"Start":"11:05.560 ","End":"11:12.370","Text":"This would be the formula which is just obtained by just copying this."},{"Start":"11:12.370 ","End":"11:16.675","Text":"Letting n equals 4 and c equals 10."},{"Start":"11:16.675 ","End":"11:23.215","Text":"Somehow between this formula and this formula,"},{"Start":"11:23.215 ","End":"11:25.720","Text":"we should be able to do the general case because we"},{"Start":"11:25.720 ","End":"11:28.450","Text":"have m x and we have n. We could do mx plus"},{"Start":"11:28.450 ","End":"11:34.990","Text":"n. Here\u0027s our example for the general short case."},{"Start":"11:34.990 ","End":"11:37.135","Text":"We\u0027re still in short case 3."},{"Start":"11:37.135 ","End":"11:41.360","Text":"This time we have x\u0027s and numbers."},{"Start":"11:41.360 ","End":"11:46.035","Text":"What I would do would be to split this up into 2 bits."},{"Start":"11:46.035 ","End":"11:51.510","Text":"We can say that this is equal to 10 times the"},{"Start":"11:51.510 ","End":"11:59.090","Text":"integral of x over x squared plus 4,"},{"Start":"11:59.220 ","End":"12:08.845","Text":"plus 11 times the integral of 1 over x squared plus 4."},{"Start":"12:08.845 ","End":"12:13.705","Text":"Each of these types is familiar to us."},{"Start":"12:13.705 ","End":"12:18.160","Text":"This 1 becomes 1/2,"},{"Start":"12:18.160 ","End":"12:21.100","Text":"which makes the 1/2 goes with the 10."},{"Start":"12:21.100 ","End":"12:27.985","Text":"I\u0027ll make it 10 over 2 times the natural log"},{"Start":"12:27.985 ","End":"12:32.005","Text":"of x squared plus 4."},{"Start":"12:32.005 ","End":"12:36.355","Text":"This would be 11 times,"},{"Start":"12:36.355 ","End":"12:45.150","Text":"or rather 11 over the square root of 4 times arctangent"},{"Start":"12:45.150 ","End":"12:54.460","Text":"of x over the square root of 4 plus constant."},{"Start":"12:54.460 ","End":"12:57.685","Text":"You see the general short case,"},{"Start":"12:57.685 ","End":"13:01.210","Text":"we just break up the m x plus n into"},{"Start":"13:01.210 ","End":"13:05.695","Text":"the case of the x\u0027s in the case of the numbers separately."},{"Start":"13:05.695 ","End":"13:08.680","Text":"This 1, we know how to do."},{"Start":"13:08.680 ","End":"13:10.705","Text":"It involves a natural logarithm."},{"Start":"13:10.705 ","End":"13:14.650","Text":"This 1 we also know how to do with the help of the arctangent."},{"Start":"13:14.650 ","End":"13:18.370","Text":"I\u0027ll just straighten up a bit here and write the final answer"},{"Start":"13:18.370 ","End":"13:22.780","Text":"as 5 times the natural log of x squared plus"},{"Start":"13:22.780 ","End":"13:32.560","Text":"4 plus 11 over 2 arctangent of x over 2."},{"Start":"13:32.560 ","End":"13:37.870","Text":"That\u0027s it. We now know how to do the short case 3."},{"Start":"13:37.870 ","End":"13:42.700","Text":"The 1 where there\u0027s just no bx in the denominator."},{"Start":"13:42.700 ","End":"13:47.095","Text":"Now we\u0027ll move on to the general case 3."},{"Start":"13:47.095 ","End":"13:52.270","Text":"We\u0027ll do the long case 3 by means of an example."},{"Start":"13:52.270 ","End":"14:00.250","Text":"Now somehow we\u0027re going to use a trick to get it to the short case 3."},{"Start":"14:00.250 ","End":"14:04.960","Text":"The trick I mentioned is something that belongs to algebra,"},{"Start":"14:04.960 ","End":"14:07.555","Text":"and it\u0027s called completing the square."},{"Start":"14:07.555 ","End":"14:09.820","Text":"You may or may not have heard of it."},{"Start":"14:09.820 ","End":"14:11.770","Text":"It\u0027s a technique, for example,"},{"Start":"14:11.770 ","End":"14:15.400","Text":"that\u0027s used in solving quadratic equations without the formula."},{"Start":"14:15.400 ","End":"14:16.735","Text":"But never mind that."},{"Start":"14:16.735 ","End":"14:18.849","Text":"Let\u0027s start off with an example."},{"Start":"14:18.849 ","End":"14:22.465","Text":"Suppose I have an expression,"},{"Start":"14:22.465 ","End":"14:26.170","Text":"a quadratic expression without the free term,"},{"Start":"14:26.170 ","End":"14:29.960","Text":"like x squared plus 6x."},{"Start":"14:30.030 ","End":"14:38.155","Text":"The idea of this trick is to complete it with a constant here and make it"},{"Start":"14:38.155 ","End":"14:47.060","Text":"equal to x plus something squared and then plus or minus something."},{"Start":"14:47.100 ","End":"14:50.830","Text":"Here it goes. If I have x plus something squared,"},{"Start":"14:50.830 ","End":"14:54.730","Text":"it\u0027s going to be x squared plus twice that something."},{"Start":"14:54.730 ","End":"14:57.510","Text":"Twice that something has to be 6x."},{"Start":"14:57.510 ","End":"14:59.600","Text":"I put 3 here."},{"Start":"14:59.600 ","End":"15:03.015","Text":"Now, if I expand this,"},{"Start":"15:03.015 ","End":"15:08.220","Text":"I get x squared plus 6x plus 9."},{"Start":"15:09.330 ","End":"15:14.650","Text":"What I have to do is say that x squared plus"},{"Start":"15:14.650 ","End":"15:20.755","Text":"6x is this thing, but minus 9."},{"Start":"15:20.755 ","End":"15:23.110","Text":"I\u0027ll show you some other examples."},{"Start":"15:23.110 ","End":"15:28.660","Text":"Suppose I had x squared minus 4x."},{"Start":"15:28.660 ","End":"15:32.635","Text":"I would write this as equaling."},{"Start":"15:32.635 ","End":"15:35.770","Text":"Start off with x something squared."},{"Start":"15:35.770 ","End":"15:37.360","Text":"This time because it\u0027s a minus,"},{"Start":"15:37.360 ","End":"15:38.575","Text":"it will be a minus."},{"Start":"15:38.575 ","End":"15:43.015","Text":"Again, 1/2 this coefficient would mean that it\u0027s minus 2."},{"Start":"15:43.015 ","End":"15:44.335","Text":"If I square this,"},{"Start":"15:44.335 ","End":"15:49.735","Text":"I get x squared minus 4x plus 4."},{"Start":"15:49.735 ","End":"15:54.385","Text":"I have to subtract 4 to make it equal."},{"Start":"15:54.385 ","End":"15:57.025","Text":"Let\u0027s take the example here."},{"Start":"15:57.025 ","End":"15:59.600","Text":"X squared plus 4x."},{"Start":"16:01.650 ","End":"16:04.765","Text":"This will be equal."},{"Start":"16:04.765 ","End":"16:08.065","Text":"This time x can you guess already,"},{"Start":"16:08.065 ","End":"16:11.155","Text":"plus 2 1/2 of this squared,"},{"Start":"16:11.155 ","End":"16:15.020","Text":"and then it\u0027s always minus distinct squared."},{"Start":"16:15.570 ","End":"16:18.985","Text":"Take an example that\u0027s not a whole number here."},{"Start":"16:18.985 ","End":"16:25.075","Text":"Suppose I had x squared plus 3x,"},{"Start":"16:25.075 ","End":"16:31.960","Text":"then this would equal x plus 3 over 2 is just 3 over 2,"},{"Start":"16:31.960 ","End":"16:34.100","Text":"or 1 and 1/2."},{"Start":"16:34.530 ","End":"16:40.630","Text":"I have to subtract something because this is x squared plus twice 3 over 2,"},{"Start":"16:40.630 ","End":"16:44.020","Text":"which is 3x plus 3 over 2 squared."},{"Start":"16:44.020 ","End":"16:50.635","Text":"I have to subtract 3 over 2 squared or 9 over 4."},{"Start":"16:50.635 ","End":"16:55.765","Text":"This is essentially the technique of completing the square."},{"Start":"16:55.765 ","End":"17:01.825","Text":"Now I\u0027m going to apply this technique to our case."},{"Start":"17:01.825 ","End":"17:05.830","Text":"What we get is the integral."},{"Start":"17:05.830 ","End":"17:09.200","Text":"Numerator, we don\u0027t touch."},{"Start":"17:09.780 ","End":"17:15.280","Text":"This part here, we use completing the square on."},{"Start":"17:15.280 ","End":"17:18.100","Text":"We even have it here, this example."},{"Start":"17:18.100 ","End":"17:24.910","Text":"It\u0027s x plus 2 squared minus 4,"},{"Start":"17:24.910 ","End":"17:27.790","Text":"but there\u0027s still a plus 10 here."},{"Start":"17:27.790 ","End":"17:32.410","Text":"Ultimately if we just combine the minus 4 and the 10,"},{"Start":"17:32.410 ","End":"17:36.550","Text":"we get the integral of x plus 1"},{"Start":"17:36.550 ","End":"17:43.910","Text":"over x plus 2 squared plus 6 dx."},{"Start":"17:44.040 ","End":"17:46.765","Text":"This is very good."},{"Start":"17:46.765 ","End":"17:52.014","Text":"We wanted to get rid of this for x because this middle term, the bx,"},{"Start":"17:52.014 ","End":"17:59.695","Text":"was what was interfering with it being a short case 3 and now it\u0027s a long case 3."},{"Start":"17:59.695 ","End":"18:01.720","Text":"That\u0027s true. This is not x squared."},{"Start":"18:01.720 ","End":"18:06.745","Text":"It\u0027s x plus 2 squared but a substitution will solve that for us."},{"Start":"18:06.745 ","End":"18:08.770","Text":"After we do completing the square,"},{"Start":"18:08.770 ","End":"18:10.420","Text":"we do a substitution."},{"Start":"18:10.420 ","End":"18:15.685","Text":"In this case, we let x plus 2 be t, for example."},{"Start":"18:15.685 ","End":"18:18.670","Text":"Our favorite letter for substitution."},{"Start":"18:18.670 ","End":"18:23.770","Text":"Dx is just equal to d t. In case we need it,"},{"Start":"18:23.770 ","End":"18:31.570","Text":"let\u0027s put x in terms of t. X will equal t minus 2 in case we need the other way."},{"Start":"18:31.570 ","End":"18:35.110","Text":"This is the substitution that we do."},{"Start":"18:35.110 ","End":"18:40.900","Text":"Now this thing becomes the integral."},{"Start":"18:40.900 ","End":"18:43.555","Text":"Let\u0027s do the denominator first."},{"Start":"18:43.555 ","End":"18:48.535","Text":"X plus 2 is t. It\u0027s t squared plus 6."},{"Start":"18:48.535 ","End":"18:57.160","Text":"Dx is dt and x plus 1 is t minus 2 plus 1."},{"Start":"18:57.160 ","End":"19:05.020","Text":"Allow me to skip a step and say t minus 2 plus 1 is t minus 1. Very well."},{"Start":"19:05.020 ","End":"19:11.365","Text":"Now we do have a k3 short type of integral."},{"Start":"19:11.365 ","End":"19:14.065","Text":"We just continue like we learned before."},{"Start":"19:14.065 ","End":"19:17.290","Text":"We split it up into 2 integrals."},{"Start":"19:17.290 ","End":"19:22.915","Text":"We write that this is equal to the integral of"},{"Start":"19:22.915 ","End":"19:30.110","Text":"t over t squared plus 6dt."},{"Start":"19:31.740 ","End":"19:33.790","Text":"There\u0027s a minus here,"},{"Start":"19:33.790 ","End":"19:41.110","Text":"so minus the integral of 1 over"},{"Start":"19:41.110 ","End":"19:48.985","Text":"t squared plus 6dt."},{"Start":"19:48.985 ","End":"19:52.555","Text":"We know how to solve each of these integrals."},{"Start":"19:52.555 ","End":"19:57.475","Text":"Here we had the thing with the natural logarithm and here the thing with the arctangent."},{"Start":"19:57.475 ","End":"19:59.290","Text":"Just to remind you,"},{"Start":"19:59.290 ","End":"20:01.240","Text":"that you have the formula. What we did here."},{"Start":"20:01.240 ","End":"20:03.025","Text":"We have to make sure it\u0027s a 2 here."},{"Start":"20:03.025 ","End":"20:06.085","Text":"We get the derivative of the denominator."},{"Start":"20:06.085 ","End":"20:07.900","Text":"We put a 2 in here,"},{"Start":"20:07.900 ","End":"20:11.255","Text":"but we also compensate by putting a 1/2 outside."},{"Start":"20:11.255 ","End":"20:17.230","Text":"What we get, is 1/2, the natural logarithm."},{"Start":"20:17.230 ","End":"20:19.360","Text":"I always write the absolute value,"},{"Start":"20:19.360 ","End":"20:23.125","Text":"although it\u0027s not necessary here because the denominator is always positive,"},{"Start":"20:23.125 ","End":"20:25.120","Text":"but just out of habit,"},{"Start":"20:25.120 ","End":"20:28.730","Text":"t squared plus 6."},{"Start":"20:29.130 ","End":"20:33.895","Text":"Here we have 1 over;"},{"Start":"20:33.895 ","End":"20:36.655","Text":"we take always the square root of this thing."},{"Start":"20:36.655 ","End":"20:38.695","Text":"So it\u0027s the square root of 6."},{"Start":"20:38.695 ","End":"20:46.570","Text":"Then the arc tangent of t over the square root of"},{"Start":"20:46.570 ","End":"20:50.790","Text":"6 plus the constant K. At this point we\u0027re"},{"Start":"20:50.790 ","End":"20:55.160","Text":"not quite done because we still have to reverse the substitution."},{"Start":"20:55.160 ","End":"20:59.990","Text":"We get to go from t back to x using this line here."},{"Start":"20:59.990 ","End":"21:10.520","Text":"What we get is 1/2 natural log of t squared is x plus 2"},{"Start":"21:10.520 ","End":"21:14.540","Text":"squared and plus"},{"Start":"21:14.540 ","End":"21:20.780","Text":"6 and minus 1"},{"Start":"21:20.780 ","End":"21:25.880","Text":"over root 6 arctangent"},{"Start":"21:25.880 ","End":"21:35.310","Text":"of x plus 2 over root 6."},{"Start":"21:35.550 ","End":"21:39.340","Text":"The last thing you could do if you wanted to would be to"},{"Start":"21:39.340 ","End":"21:43.165","Text":"replace this by what it is expanded."},{"Start":"21:43.165 ","End":"21:49.225","Text":"If you think about it, it\u0027s x squared plus 4x plus 4 plus 6 and that is simply"},{"Start":"21:49.225 ","End":"21:55.855","Text":"x squared plus 4x plus 10 if we want to do that."},{"Start":"21:55.855 ","End":"22:01.180","Text":"Because then you see what this x squared plus 4 x plus 10 is,"},{"Start":"22:01.180 ","End":"22:03.985","Text":"it\u0027s just the original denominator."},{"Start":"22:03.985 ","End":"22:06.715","Text":"Anyway, we\u0027re done with this."},{"Start":"22:06.715 ","End":"22:10.345","Text":"Let\u0027s go and do another example,"},{"Start":"22:10.345 ","End":"22:12.835","Text":"but before I go on to the next example,"},{"Start":"22:12.835 ","End":"22:16.345","Text":"I would like to do 1 more example of completing the square."},{"Start":"22:16.345 ","End":"22:19.180","Text":"The reason I want to do that is that this is what I\u0027m going to get in"},{"Start":"22:19.180 ","End":"22:26.575","Text":"the next exercise and I\u0027d like to complete the square for x squared plus x."},{"Start":"22:26.575 ","End":"22:31.120","Text":"This time it\u0027s going to equal. Remember how we do this?"},{"Start":"22:31.120 ","End":"22:33.130","Text":"We take 0.5 of this coefficient,"},{"Start":"22:33.130 ","End":"22:36.860","Text":"which is 1, and make it plus 0.5."},{"Start":"22:36.900 ","End":"22:39.085","Text":"After I square this,"},{"Start":"22:39.085 ","End":"22:42.100","Text":"I am going to get x squared plus twice x times 0.5,"},{"Start":"22:42.100 ","End":"22:46.420","Text":"which is plus x, but the last term a constant will be 0.25."},{"Start":"22:46.420 ","End":"22:53.965","Text":"I have to remove that by subtracting 0.5 squared or if you like, 0.25."},{"Start":"22:53.965 ","End":"22:59.260","Text":"This 1 would be instead minus 3 over 2,"},{"Start":"22:59.260 ","End":"23:03.820","Text":"I could write minus 9 over 4 and instead of minus 0.5 squared,"},{"Start":"23:03.820 ","End":"23:06.175","Text":"I could do minus 0.25."},{"Start":"23:06.175 ","End":"23:11.570","Text":"Now, this 1 I\u0027m going to use in the next exercise."},{"Start":"23:11.670 ","End":"23:15.415","Text":"Here we are with case 3,"},{"Start":"23:15.415 ","End":"23:19.689","Text":"the long type another example,"},{"Start":"23:19.689 ","End":"23:24.080","Text":"actually this is going to be the last example."},{"Start":"23:24.420 ","End":"23:30.265","Text":"Here we are, 1 over x squared plus x plus 1."},{"Start":"23:30.265 ","End":"23:35.185","Text":"If you check it, you\u0027ll find you have no solutions to this equation."},{"Start":"23:35.185 ","End":"23:39.265","Text":"You either try solving it or you could check the discriminant"},{"Start":"23:39.265 ","End":"23:43.885","Text":"b squared minus 4ac and find that it\u0027s negative."},{"Start":"23:43.885 ","End":"23:45.835","Text":"I won\u0027t do that,"},{"Start":"23:45.835 ","End":"23:51.325","Text":"but I will remember that on the previous page,"},{"Start":"23:51.325 ","End":"23:56.335","Text":"we found out by completing the square that x squared plus"},{"Start":"23:56.335 ","End":"24:02.905","Text":"x is equal to x plus 1/2 squared minus 1/4."},{"Start":"24:02.905 ","End":"24:05.020","Text":"I\u0027m going to use that here."},{"Start":"24:05.020 ","End":"24:13.015","Text":"That\u0027s the first thing I do when I have a term of bx here."},{"Start":"24:13.015 ","End":"24:16.884","Text":"I don\u0027t want to get rid of that,"},{"Start":"24:16.884 ","End":"24:19.525","Text":"because if I didn\u0027t have this,"},{"Start":"24:19.525 ","End":"24:21.250","Text":"then I would have the short case,"},{"Start":"24:21.250 ","End":"24:23.110","Text":"which is what I want."},{"Start":"24:23.110 ","End":"24:27.370","Text":"This completing the square will help get rid of this middle term."},{"Start":"24:27.370 ","End":"24:34.960","Text":"I continue the integral of 1 over, from here,"},{"Start":"24:34.960 ","End":"24:42.890","Text":"it\u0027s x plus 0.5 all squared minus a quarter,"},{"Start":"24:43.710 ","End":"24:50.080","Text":"but I still have a plus 1 here, dx."},{"Start":"24:50.080 ","End":"24:53.050","Text":"That means that, just a slight rewrite,"},{"Start":"24:53.050 ","End":"24:57.460","Text":"it\u0027s 1 over x plus"},{"Start":"24:57.460 ","End":"25:06.100","Text":"1/2 squared plus 3/4."},{"Start":"25:06.100 ","End":"25:09.715","Text":"From here, you can also see why the denominator is never 0."},{"Start":"25:09.715 ","End":"25:12.520","Text":"It\u0027s something squared plus something positive,"},{"Start":"25:12.520 ","End":"25:16.780","Text":"so it has to be positive and so this will never be 0."},{"Start":"25:16.780 ","End":"25:21.070","Text":"At this stage, what we do is a substitution."},{"Start":"25:21.070 ","End":"25:24.535","Text":"We let x plus 0.5 equal t,"},{"Start":"25:24.535 ","End":"25:28.070","Text":"dx is just equal to dt."},{"Start":"25:28.170 ","End":"25:31.540","Text":"When we do the reverse substitution,"},{"Start":"25:31.540 ","End":"25:36.730","Text":"we can also say that x in terms of t is"},{"Start":"25:36.730 ","End":"25:42.325","Text":"t minus 0.5 and with this,"},{"Start":"25:42.325 ","End":"25:43.945","Text":"we can now continue."},{"Start":"25:43.945 ","End":"25:45.955","Text":"We get the integral,"},{"Start":"25:45.955 ","End":"25:50.930","Text":"and now an integral in terms of t. It\u0027s 1 over"},{"Start":"25:50.970 ","End":"25:58.180","Text":"t squared plus 3/4 dt."},{"Start":"25:58.180 ","End":"26:04.000","Text":"Now, this we already know how to do in terms of the arc tangent."},{"Start":"26:04.000 ","End":"26:10.960","Text":"This is just equal to 1 over the square root of 3/4 times"},{"Start":"26:10.960 ","End":"26:16.915","Text":"the arctangent of t"},{"Start":"26:16.915 ","End":"26:25.120","Text":"over the square root of 0.75 plus constant."},{"Start":"26:25.120 ","End":"26:31.615","Text":"We\u0027d just like to rewrite the square root of 0.75 and I\u0027ll do that at the side,"},{"Start":"26:31.615 ","End":"26:38.530","Text":"1 over the square root of 0.75 is equal to."},{"Start":"26:38.530 ","End":"26:41.155","Text":"Because it\u0027s on the denominator,"},{"Start":"26:41.155 ","End":"26:45.160","Text":"I can reverse the numerator and denominator here and"},{"Start":"26:45.160 ","End":"26:50.140","Text":"make it equal to the square root of 4 over 3."},{"Start":"26:50.140 ","End":"26:51.984","Text":"This is really 2 steps,"},{"Start":"26:51.984 ","End":"26:57.190","Text":"but you can probably see this and then the square root of 4"},{"Start":"26:57.190 ","End":"27:03.295","Text":"happens to come out a whole number so I can write this as 2 over square root of 3."},{"Start":"27:03.295 ","End":"27:05.515","Text":"Anyway, this is optional,"},{"Start":"27:05.515 ","End":"27:07.360","Text":"could leave it like this."},{"Start":"27:07.360 ","End":"27:14.680","Text":"I just like to tidy up a bit, there we are."},{"Start":"27:14.680 ","End":"27:17.050","Text":"The next step, if you remember,"},{"Start":"27:17.050 ","End":"27:20.720","Text":"is going from t back to x."},{"Start":"27:23.310 ","End":"27:30.055","Text":"When we see t, we put x plus 1/2, so we get."},{"Start":"27:30.055 ","End":"27:34.720","Text":"Now, this is 2 over the square root of 3."},{"Start":"27:34.720 ","End":"27:38.200","Text":"We did over the side, or you can leave it like this."},{"Start":"27:38.200 ","End":"27:44.560","Text":"Arc tangent of 2 over square root of 3"},{"Start":"27:44.560 ","End":"27:51.650","Text":"makes this 2t over the square root of 3 plus a constant."},{"Start":"27:51.660 ","End":"27:55.285","Text":"I haven\u0027t yet substituted the t, yeah, sorry."},{"Start":"27:55.285 ","End":"28:05.530","Text":"It\u0027s 2 over square root of 3 arc tangent of twice x"},{"Start":"28:05.530 ","End":"28:12.310","Text":"plus 1/2 over square root of 3 plus"},{"Start":"28:12.310 ","End":"28:21.055","Text":"k. For those of you who don\u0027t like to mess around with square roots and so forth,"},{"Start":"28:21.055 ","End":"28:26.455","Text":"I\u0027ll write the answer also in the original form as if I had continued from here."},{"Start":"28:26.455 ","End":"28:31.345","Text":"It will be 1 over the square root of"},{"Start":"28:31.345 ","End":"28:37.390","Text":"3 over 4 times arc tangent"},{"Start":"28:37.390 ","End":"28:43.240","Text":"of x plus 0.5 over"},{"Start":"28:43.240 ","End":"28:51.760","Text":"the same square root of 3 over 4 plus the constant."},{"Start":"28:51.760 ","End":"28:56.875","Text":"Either 1 of these forms for those who like simplifying and for those who don\u0027t."},{"Start":"28:56.875 ","End":"29:01.540","Text":"In fact, those who like simplifying could even go 1 step further and"},{"Start":"29:01.540 ","End":"29:07.255","Text":"write this numerator here as 2x plus 1,"},{"Start":"29:07.255 ","End":"29:10.010","Text":"for those who like that sort of thing."},{"Start":"29:11.160 ","End":"29:13.510","Text":"We\u0027re done with this example,"},{"Start":"29:13.510 ","End":"29:15.010","Text":"we\u0027re done with case 3,"},{"Start":"29:15.010 ","End":"29:22.580","Text":"and we\u0027re done with all 3 cases of the basic case for rational functions."}],"ID":4493},{"Watched":false,"Name":"A Note on the Coefficient of X-Squared","Duration":"1m 58s","ChapterTopicVideoID":4478,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.680","Text":"A note on the coefficient of x squared,"},{"Start":"00:02.680 ","End":"00:04.590","Text":"and I\u0027m still talking about"},{"Start":"00:04.590 ","End":"00:07.665","Text":"the basic case of rational functions."},{"Start":"00:07.665 ","End":"00:09.720","Text":"Up to now, we\u0027ve been dealing"},{"Start":"00:09.720 ","End":"00:14.100","Text":"with the integral of the type mx plus n"},{"Start":"00:14.100 ","End":"00:19.950","Text":"over x squared plus bx plus c."},{"Start":"00:19.950 ","End":"00:21.345","Text":"Just to emphasize it,"},{"Start":"00:21.345 ","End":"00:23.310","Text":"there is no coefficient here apparently,"},{"Start":"00:23.310 ","End":"00:25.080","Text":"but the coefficient is 1."},{"Start":"00:25.080 ","End":"00:27.750","Text":"This seems like an unnecessary restriction."},{"Start":"00:27.750 ","End":"00:29.849","Text":"After all, if we\u0027ve got an integral"},{"Start":"00:29.849 ","End":"00:32.325","Text":"such as this for example,"},{"Start":"00:32.325 ","End":"00:35.475","Text":"where the coefficient of x squared is not 1,"},{"Start":"00:35.475 ","End":"00:38.045","Text":"it\u0027s 2, what do we do then?"},{"Start":"00:38.045 ","End":"00:40.970","Text":"In this case where the coefficient is not 1,"},{"Start":"00:40.970 ","End":"00:43.205","Text":"then the trick, it\u0027s not really a trick,"},{"Start":"00:43.205 ","End":"00:45.980","Text":"is to take the coefficient outside the integral."},{"Start":"00:45.980 ","End":"00:47.675","Text":"Now, what do I mean by this?"},{"Start":"00:47.675 ","End":"00:49.900","Text":"I simply mean that if we take 2"},{"Start":"00:49.900 ","End":"00:52.145","Text":"out of the brackets algebraically,"},{"Start":"00:52.145 ","End":"00:53.270","Text":"and then we can take it out"},{"Start":"00:53.270 ","End":"00:55.400","Text":"altogether outside the integral,"},{"Start":"00:55.400 ","End":"00:58.250","Text":"what we\u0027re left with is 1 over 2,"},{"Start":"00:58.250 ","End":"01:00.590","Text":"the 2 from here, times the integral."},{"Start":"01:00.590 ","End":"01:02.400","Text":"Numerator is the same."},{"Start":"01:02.400 ","End":"01:03.779","Text":"But from the denominator,"},{"Start":"01:03.779 ","End":"01:05.055","Text":"we take 2 out."},{"Start":"01:05.055 ","End":"01:07.290","Text":"The simplest way to do this practically,"},{"Start":"01:07.290 ","End":"01:10.040","Text":"is just to divide each of the coefficients by 2,"},{"Start":"01:10.040 ","End":"01:13.085","Text":"so if we had 2, now we only have 1."},{"Start":"01:13.085 ","End":"01:15.125","Text":"Here, we have the coefficient of 1,"},{"Start":"01:15.125 ","End":"01:18.270","Text":"so we now have a coefficient of 1/2."},{"Start":"01:18.270 ","End":"01:21.900","Text":"Here, we have plus 1, so it becomes 1/2."},{"Start":"01:21.900 ","End":"01:24.845","Text":"If you\u0027re still not clear that this is right,"},{"Start":"01:24.845 ","End":"01:27.485","Text":"just multiply 2 by this thing,"},{"Start":"01:27.485 ","End":"01:29.810","Text":"and you\u0027ll see that we get back to this."},{"Start":"01:29.810 ","End":"01:32.450","Text":"Once we get to this integral,"},{"Start":"01:32.450 ","End":"01:35.115","Text":"we then proceed as usual."},{"Start":"01:35.115 ","End":"01:37.080","Text":"That\u0027s basically what it is,"},{"Start":"01:37.080 ","End":"01:39.470","Text":"and we\u0027ve generalized it now to any"},{"Start":"01:39.470 ","End":"01:42.545","Text":"degree 1 over degree 2 rational function."},{"Start":"01:42.545 ","End":"01:44.090","Text":"In general, if it wasn\u0027t 2"},{"Start":"01:44.090 ","End":"01:45.170","Text":"or some other number,"},{"Start":"01:45.170 ","End":"01:46.920","Text":"let\u0027s say it was 40 here,"},{"Start":"01:46.920 ","End":"01:49.730","Text":"we take 1 over 40 in front of the integral"},{"Start":"01:49.730 ","End":"01:51.360","Text":"and divide each thing by 40,"},{"Start":"01:51.360 ","End":"01:54.245","Text":"and then we\u0027ll be left with x squared on its own,"},{"Start":"01:54.245 ","End":"01:55.970","Text":"like a coefficient of 1."},{"Start":"01:55.970 ","End":"01:58.860","Text":"That\u0027s all I need to say about this."}],"ID":4494},{"Watched":false,"Name":"General Case - First Step","Duration":"6m 47s","ChapterTopicVideoID":4485,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"Now we\u0027re going to talk about the general case of a rational function."},{"Start":"00:03.900 ","End":"00:07.950","Text":"Up until now, we\u0027ve dealt mostly with linear over a quadratic,"},{"Start":"00:07.950 ","End":"00:09.675","Text":"what we call the basic case,"},{"Start":"00:09.675 ","End":"00:18.255","Text":"let\u0027s say mx plus n over x squared plus bx plus c integral of that,"},{"Start":"00:18.255 ","End":"00:20.850","Text":"and also some immediate integrals."},{"Start":"00:20.850 ","End":"00:26.505","Text":"For example, the integral of 1 over x plus a and that sort of thing."},{"Start":"00:26.505 ","End":"00:30.135","Text":"We even generalized this a bit where it\u0027s not just 1 as a coefficient,"},{"Start":"00:30.135 ","End":"00:34.410","Text":"but we want to generalize much more to almost every rational function."},{"Start":"00:34.410 ","End":"00:38.320","Text":"I copied this stuff from the introduction clip."},{"Start":"00:38.320 ","End":"00:42.650","Text":"This was the definition of a rational function and here were a bunch of example."},{"Start":"00:42.650 ","End":"00:46.070","Text":"I\u0027m going to generalize to the case where"},{"Start":"00:46.070 ","End":"00:50.285","Text":"the degree in the denominator is bigger than the degree in the numerator,"},{"Start":"00:50.285 ","End":"00:56.780","Text":"so that what I mean is that n is less than m. Later on,"},{"Start":"00:56.780 ","End":"00:59.045","Text":"we\u0027ll see that this is not even a restriction,"},{"Start":"00:59.045 ","End":"01:01.790","Text":"but we need a technique called division of"},{"Start":"01:01.790 ","End":"01:06.290","Text":"polynomials or long division of polynomials in order to get the fully general case."},{"Start":"01:06.290 ","End":"01:09.650","Text":"This time we\u0027ll restrict ourselves to this condition,"},{"Start":"01:09.650 ","End":"01:12.710","Text":"and if I write the degrees of these polynomials,"},{"Start":"01:12.710 ","End":"01:15.020","Text":"this degree, for example,"},{"Start":"01:15.020 ","End":"01:18.260","Text":"and the numerator is 4 and the denominator is 2."},{"Start":"01:18.260 ","End":"01:22.415","Text":"Here we have 3, and 2 and 2 is 4,"},{"Start":"01:22.415 ","End":"01:25.609","Text":"degree 2, degree 2,"},{"Start":"01:25.609 ","End":"01:29.325","Text":"and degree 1, degree 12."},{"Start":"01:29.325 ","End":"01:31.129","Text":"Out of all of these,"},{"Start":"01:31.129 ","End":"01:34.850","Text":"the ones that we will be able to do is like 3, 4, 0, 2."},{"Start":"01:34.850 ","End":"01:36.595","Text":"The top is lower than the bottom."},{"Start":"01:36.595 ","End":"01:40.070","Text":"This one I won\u0027t be able to do just yet"},{"Start":"01:40.070 ","End":"01:43.970","Text":"and won\u0027t be able to do this one because it\u0027s 2 over 2,"},{"Start":"01:43.970 ","End":"01:45.410","Text":"even equality is no good;"},{"Start":"01:45.410 ","End":"01:47.510","Text":"n has to be strictly less than m,"},{"Start":"01:47.510 ","End":"01:49.970","Text":"degree lower in the top than in the bottom."},{"Start":"01:49.970 ","End":"01:52.315","Text":"This one we also won\u0027t be able to do."},{"Start":"01:52.315 ","End":"01:55.350","Text":"I\u0027ll just mention and in the case it\u0027s not this either,"},{"Start":"01:55.350 ","End":"01:57.195","Text":"n is bigger or equal to m,"},{"Start":"01:57.195 ","End":"02:00.680","Text":"we need to do something called polynomial division."},{"Start":"02:00.680 ","End":"02:03.154","Text":"We\u0027ll learn this later."},{"Start":"02:03.154 ","End":"02:08.975","Text":"The first step will be to factorize the denominator and I\u0027ll show you some example."},{"Start":"02:08.975 ","End":"02:12.705","Text":"All I\u0027m doing is factorizing the denominator."},{"Start":"02:12.705 ","End":"02:15.380","Text":"In this first case, it\u0027s fairly easy."},{"Start":"02:15.380 ","End":"02:18.829","Text":"For example, we could take x outside the brackets,"},{"Start":"02:18.829 ","End":"02:24.440","Text":"get x times x squared minus 1 and this goes to x minus 1,"},{"Start":"02:24.440 ","End":"02:27.949","Text":"x plus 1, so we end up by getting"},{"Start":"02:27.949 ","End":"02:33.290","Text":"this and numerator I just left alone and I factorize the denominator."},{"Start":"02:33.290 ","End":"02:40.010","Text":"In this case, the denominator factors into x plus 1,"},{"Start":"02:40.010 ","End":"02:43.909","Text":"x plus 2, x minus 3."},{"Start":"02:43.909 ","End":"02:46.325","Text":"I won\u0027t tell you exactly how I did it because"},{"Start":"02:46.325 ","End":"02:50.165","Text":"the art of factorizing is a thing in itself."},{"Start":"02:50.165 ","End":"02:53.735","Text":"You can check it by multiplying this out and see that we get this."},{"Start":"02:53.735 ","End":"02:59.795","Text":"Keeping the numerator the same as 6x squared plus 4x minus 6. We\u0027ll continue."},{"Start":"02:59.795 ","End":"03:04.700","Text":"This one also, I\u0027ll just tell you the answer without showing you how I did it,"},{"Start":"03:04.700 ","End":"03:08.780","Text":"I will just tell you in general that we use a substitution to t"},{"Start":"03:08.780 ","End":"03:14.025","Text":"equals x squared then we get t minus 9,"},{"Start":"03:14.025 ","End":"03:19.320","Text":"t minus 4, that\u0027s x squared minus 9 x squared minus 4."},{"Start":"03:19.320 ","End":"03:21.210","Text":"From the x squared minus 9,"},{"Start":"03:21.210 ","End":"03:26.160","Text":"we get x plus 3, x minus 3."},{"Start":"03:26.160 ","End":"03:28.575","Text":"From the x squared minus 4,"},{"Start":"03:28.575 ","End":"03:31.185","Text":"we get x plus 2,"},{"Start":"03:31.185 ","End":"03:35.120","Text":"x minus 2 dx, and the numerator,"},{"Start":"03:35.120 ","End":"03:38.299","Text":"just as it was, 10 x."},{"Start":"03:38.299 ","End":"03:42.680","Text":"In this case, we can take x outside the brackets and"},{"Start":"03:42.680 ","End":"03:46.670","Text":"then we\u0027re left with x squared plus 6x plus 9,"},{"Start":"03:46.670 ","End":"03:48.380","Text":"which is x plus 3 squared."},{"Start":"03:48.380 ","End":"03:50.150","Text":"What\u0027s on top is the same,"},{"Start":"03:50.150 ","End":"03:54.950","Text":"9x plus 36 dx integral."},{"Start":"03:54.950 ","End":"03:58.610","Text":"This is just x minus 1 squared,"},{"Start":"03:58.610 ","End":"04:00.800","Text":"1 over x minus 1 squared,"},{"Start":"04:00.800 ","End":"04:02.270","Text":"and this is also a perfect square,"},{"Start":"04:02.270 ","End":"04:05.020","Text":"x minus 2 squared dx."},{"Start":"04:05.020 ","End":"04:07.610","Text":"I\u0027m just giving it a general idea that we"},{"Start":"04:07.610 ","End":"04:10.790","Text":"factorize the denominator and this is just so important."},{"Start":"04:10.790 ","End":"04:12.605","Text":"Let\u0027s do a few more examples."},{"Start":"04:12.605 ","End":"04:14.225","Text":"Here\u0027s what I get."},{"Start":"04:14.225 ","End":"04:18.080","Text":"Each one of these denominators I\u0027ve factorized and if we"},{"Start":"04:18.080 ","End":"04:22.100","Text":"look at all the factorized polynomials,"},{"Start":"04:22.100 ","End":"04:27.140","Text":"what you\u0027ll notice is that there\u0027s only 2 kinds of factor basically."},{"Start":"04:27.140 ","End":"04:31.190","Text":"We either have an x plus or minus a number, like here,"},{"Start":"04:31.190 ","End":"04:34.670","Text":"here, here, here, and maybe to a power like here,"},{"Start":"04:34.670 ","End":"04:35.930","Text":"to the power of 2."},{"Start":"04:35.930 ","End":"04:41.510","Text":"The other kind we get are quadratic and I\u0027ll just mark them."},{"Start":"04:41.510 ","End":"04:45.930","Text":"All of these quadratic are irreducible,"},{"Start":"04:45.930 ","End":"04:50.524","Text":"are unfactorizable means that they cannot be factored any further,"},{"Start":"04:50.524 ","End":"04:53.510","Text":"which also means that they have no roots."},{"Start":"04:53.510 ","End":"04:57.530","Text":"The equation x squared plus 1 equals 0 has no solutions."},{"Start":"04:57.530 ","End":"04:59.000","Text":"That\u0027s basically what we get."},{"Start":"04:59.000 ","End":"05:04.000","Text":"Linear and irreducible quadratics as factors."},{"Start":"05:04.000 ","End":"05:07.505","Text":"Polynomials like this and this, and this,"},{"Start":"05:07.505 ","End":"05:11.765","Text":"that can\u0027t be factorized any further are called irreducible,"},{"Start":"05:11.765 ","End":"05:14.450","Text":"and these are irreducible quadratics."},{"Start":"05:14.450 ","End":"05:19.370","Text":"I mentioned this because I want to generalize that when you have a rational function,"},{"Start":"05:19.370 ","End":"05:21.335","Text":"a polynomial over a polynomial,"},{"Start":"05:21.335 ","End":"05:27.920","Text":"it can always be brought to the form of the integral of the same p of x,"},{"Start":"05:27.920 ","End":"05:30.005","Text":"I don\u0027t do anything with the numerator,"},{"Start":"05:30.005 ","End":"05:38.570","Text":"but the denominator can always be factorized into linear factors like x minus a,"},{"Start":"05:38.570 ","End":"05:43.445","Text":"perhaps as it is or maybe to some power k and then another one,"},{"Start":"05:43.445 ","End":"05:47.720","Text":"let\u0027s say x minus b to the power of l,"},{"Start":"05:47.720 ","End":"05:49.190","Text":"and so on and so on."},{"Start":"05:49.190 ","End":"05:51.065","Text":"These are the linear terms,"},{"Start":"05:51.065 ","End":"05:56.040","Text":"and then irreducible quadratics like x squared"},{"Start":"05:56.040 ","End":"06:01.340","Text":"plus mx plus n. But on condition that this is irreducible,"},{"Start":"06:01.340 ","End":"06:05.025","Text":"doesn\u0027t have any roots to the power of i,"},{"Start":"06:05.025 ","End":"06:07.805","Text":"and so on and others like this."},{"Start":"06:07.805 ","End":"06:12.800","Text":"Basically, every polynomial can be factorized into linear things like x"},{"Start":"06:12.800 ","End":"06:18.320","Text":"minus a and irreducible quadratics of the form x squared plus mx plus n,"},{"Start":"06:18.320 ","End":"06:20.825","Text":"but on the condition that they have no roots."},{"Start":"06:20.825 ","End":"06:25.160","Text":"Let\u0027s proceed. First, I\u0027ll write that every factor in"},{"Start":"06:25.160 ","End":"06:31.135","Text":"the denominator is either linear or an irreducible quadratic."},{"Start":"06:31.135 ","End":"06:33.620","Text":"Even here, if it has an exponent, it\u0027s still,"},{"Start":"06:33.620 ","End":"06:40.100","Text":"each individual factor is either linear like x minus a or an irreducible quadratic."},{"Start":"06:40.100 ","End":"06:41.480","Text":"Instead of an exponent,"},{"Start":"06:41.480 ","End":"06:42.800","Text":"I can write it several times."},{"Start":"06:42.800 ","End":"06:44.990","Text":"Each factor is one of these or one of these;"},{"Start":"06:44.990 ","End":"06:47.940","Text":"linear or irreducible quadratic."}],"ID":4495},{"Watched":false,"Name":"General Case - Second Step","Duration":"6m 57s","ChapterTopicVideoID":4486,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"We\u0027re done with the first step which was factorizing the denominator."},{"Start":"00:04.830 ","End":"00:09.960","Text":"Now we come to this second step where partial fractions come into the picture."},{"Start":"00:09.960 ","End":"00:14.775","Text":"Well, after we\u0027ve done the factorizing of the denominator of the rational function,"},{"Start":"00:14.775 ","End":"00:20.295","Text":"it\u0027s now time to decompose the rational function into partial fractions."},{"Start":"00:20.295 ","End":"00:22.020","Text":"There are 2 main rules,"},{"Start":"00:22.020 ","End":"00:24.045","Text":"I\u0027ll write and then I\u0027ll explain."},{"Start":"00:24.045 ","End":"00:29.370","Text":"Rule number 1 says that each linear factor in the denominator of the form x"},{"Start":"00:29.370 ","End":"00:37.425","Text":"minus a^k contributes a sum of the following form as I\u0027ve written here, and I\u0027ll explain."},{"Start":"00:37.425 ","End":"00:41.460","Text":"First of all, the reason I write linear in brackets is it\u0027s not really linear,"},{"Start":"00:41.460 ","End":"00:43.515","Text":"it\u0027s linear to a power."},{"Start":"00:43.515 ","End":"00:48.130","Text":"Contributes means we have a total sum and this adds this to the total,"},{"Start":"00:48.130 ","End":"00:49.435","Text":"but this will be clear."},{"Start":"00:49.435 ","End":"00:51.160","Text":"Let\u0027s take an example."},{"Start":"00:51.160 ","End":"00:53.470","Text":"It doesn\u0027t really matter about the numerator,"},{"Start":"00:53.470 ","End":"01:02.380","Text":"could be any polynomial of x over x minus 1 squared x plus 4 cubed."},{"Start":"01:02.380 ","End":"01:08.850","Text":"What this will equal is A over x minus 1,"},{"Start":"01:08.850 ","End":"01:13.410","Text":"plus B over x minus 1 squared,"},{"Start":"01:13.410 ","End":"01:17.535","Text":"plus C over x plus 4,"},{"Start":"01:17.535 ","End":"01:21.885","Text":"plus D over x plus 4 squared,"},{"Start":"01:21.885 ","End":"01:26.915","Text":"plus E over x plus 4 cubed."},{"Start":"01:26.915 ","End":"01:28.295","Text":"Several things to note."},{"Start":"01:28.295 ","End":"01:32.630","Text":"First of all, I don\u0027t use A_1, A_2, A_3, A_4,"},{"Start":"01:32.630 ","End":"01:37.165","Text":"and so on, I prefer just to use consecutive letters of the alphabet."},{"Start":"01:37.165 ","End":"01:40.910","Text":"I write it like this because I don\u0027t know how many constants there are."},{"Start":"01:40.910 ","End":"01:44.330","Text":"Also, obviously I don\u0027t need to write to the power of 1."},{"Start":"01:44.330 ","End":"01:47.450","Text":"I think the word contributes will become clearer later."},{"Start":"01:47.450 ","End":"01:49.580","Text":"Meanwhile, let\u0027s just take another example,"},{"Start":"01:49.580 ","End":"01:55.175","Text":"where on the denominator we have x cubed times x minus 4 squared."},{"Start":"01:55.175 ","End":"01:59.445","Text":"This will contribute A over x,"},{"Start":"01:59.445 ","End":"02:02.355","Text":"plus B over x squared,"},{"Start":"02:02.355 ","End":"02:05.690","Text":"plus C over x cubed for this bit,"},{"Start":"02:05.690 ","End":"02:08.880","Text":"notice we stop at 3 3, just like here,"},{"Start":"02:08.880 ","End":"02:11.580","Text":"we stopped at power 2 up to 2,"},{"Start":"02:11.580 ","End":"02:13.695","Text":"and here, up to 3 up to 3."},{"Start":"02:13.695 ","End":"02:19.925","Text":"The next bit is if continue with the alphabet D over x minus"},{"Start":"02:19.925 ","End":"02:27.095","Text":"4^1 plus E over x minus 4 squared."},{"Start":"02:27.095 ","End":"02:30.860","Text":"Yet another example here, on the denominator,"},{"Start":"02:30.860 ","End":"02:35.330","Text":"x minus 1 times x plus 1 squared,"},{"Start":"02:35.330 ","End":"02:40.230","Text":"that will give us A over x minus 1,"},{"Start":"02:40.230 ","End":"02:41.700","Text":"that\u0027s all for this."},{"Start":"02:41.700 ","End":"02:43.890","Text":"This will give us 2 terms,"},{"Start":"02:43.890 ","End":"02:52.430","Text":"B over x plus 1 plus C over x plus 1 squared."},{"Start":"02:52.430 ","End":"02:55.410","Text":"Next, I\u0027ll go to number 2."},{"Start":"02:55.410 ","End":"02:57.390","Text":"Here it is Rule 2."},{"Start":"02:57.390 ","End":"03:02.090","Text":"Each quadratic factor, these are the irreducible quadratics, of course,"},{"Start":"03:02.090 ","End":"03:08.270","Text":"in the denominator of the form x squared plus bx plus C to some power k could be 1,"},{"Start":"03:08.270 ","End":"03:10.310","Text":"contributes a sum of the form."},{"Start":"03:10.310 ","End":"03:12.440","Text":"It\u0027s a bit different than it was before."},{"Start":"03:12.440 ","End":"03:14.780","Text":"Before we had just constants on the top,"},{"Start":"03:14.780 ","End":"03:20.405","Text":"here we have linear expressions on the top, A_1x plus B_1,"},{"Start":"03:20.405 ","End":"03:24.830","Text":"A_2 x plus B_2 over this thing squared and so on,"},{"Start":"03:24.830 ","End":"03:29.330","Text":"A_kx plus B_k over this thing to the power of"},{"Start":"03:29.330 ","End":"03:34.175","Text":"k. I should perhaps say the quadratic is not really quadratic,"},{"Start":"03:34.175 ","End":"03:37.100","Text":"it\u0027s quadratic to the power of k. Next,"},{"Start":"03:37.100 ","End":"03:40.740","Text":"I\u0027ll give a few examples to make it easier to understand."},{"Start":"03:40.740 ","End":"03:44.150","Text":"First example, where on the denominator we have"},{"Start":"03:44.150 ","End":"03:47.520","Text":"x squared plus 4 times x squared plus x plus 1,"},{"Start":"03:47.520 ","End":"03:51.745","Text":"and each 1 of them is not raised to any power or power 1."},{"Start":"03:51.745 ","End":"03:56.690","Text":"For each factor we get something like Ax plus B on the top,"},{"Start":"03:56.690 ","End":"04:02.810","Text":"so Ax plus B over x squared plus 4."},{"Start":"04:02.810 ","End":"04:08.090","Text":"Then another linear, only say Cx plus D. As before,"},{"Start":"04:08.090 ","End":"04:09.710","Text":"I don\u0027t use these subscripts,"},{"Start":"04:09.710 ","End":"04:11.540","Text":"they are just for the general formula."},{"Start":"04:11.540 ","End":"04:18.045","Text":"Cx plus D over x squared plus x plus 1."},{"Start":"04:18.045 ","End":"04:19.980","Text":"Now, let\u0027s to go for another example."},{"Start":"04:19.980 ","End":"04:23.390","Text":"As I say, the numerator doesn\u0027t really matter here,"},{"Start":"04:23.390 ","End":"04:27.475","Text":"but we know that its degree is less than the total degree of the denominator."},{"Start":"04:27.475 ","End":"04:32.915","Text":"This time, p of x over x squared plus 1 x minus 2 squared."},{"Start":"04:32.915 ","End":"04:41.714","Text":"By Rule 2, this gives us Ax plus B over x squared plus 1,"},{"Start":"04:41.714 ","End":"04:45.025","Text":"and from Rule 1, this is to the power of 2."},{"Start":"04:45.025 ","End":"04:47.315","Text":"This is just a constant,"},{"Start":"04:47.315 ","End":"04:50.360","Text":"C over x minus 2,"},{"Start":"04:50.360 ","End":"04:54.710","Text":"D over x minus 2 squared."},{"Start":"04:54.710 ","End":"04:56.990","Text":"We are up to power 2 because it\u0027s power 2 here."},{"Start":"04:56.990 ","End":"04:59.840","Text":"A third example, again,"},{"Start":"04:59.840 ","End":"05:02.885","Text":"the numerator is not really important."},{"Start":"05:02.885 ","End":"05:09.050","Text":"On the denominator here we have a quadratic squared and a linear squared."},{"Start":"05:09.050 ","End":"05:13.365","Text":"This is going to equal for this bit by Rule 2,"},{"Start":"05:13.365 ","End":"05:20.615","Text":"we\u0027ll have Ax plus B over x squared plus 4 to the power of 1 here,"},{"Start":"05:20.615 ","End":"05:26.360","Text":"and Cx plus D over x squared plus 4 squared,"},{"Start":"05:26.360 ","End":"05:27.800","Text":"we go up to power 2."},{"Start":"05:27.800 ","End":"05:30.680","Text":"Now by Rule 1 for x minus 2 squared,"},{"Start":"05:30.680 ","End":"05:37.970","Text":"we contribute E over x minus 2,"},{"Start":"05:37.970 ","End":"05:44.430","Text":"and after E comes F over x minus 2 squared."},{"Start":"05:44.430 ","End":"05:47.590","Text":"I hope this explains Rule 2."},{"Start":"05:47.590 ","End":"05:49.745","Text":"Now how do we combine all this?"},{"Start":"05:49.745 ","End":"05:54.695","Text":"Perhaps this last example explains best the concept of contributes."},{"Start":"05:54.695 ","End":"06:00.075","Text":"See this bit by Rule 2 contributes these first 2,"},{"Start":"06:00.075 ","End":"06:02.910","Text":"and this contributes these 2,"},{"Start":"06:02.910 ","End":"06:04.290","Text":"and we just add them all together."},{"Start":"06:04.290 ","End":"06:08.885","Text":"Each piece here adds more terms into this sum."},{"Start":"06:08.885 ","End":"06:11.435","Text":"The last example is the most typical."},{"Start":"06:11.435 ","End":"06:15.935","Text":"The linear factor x minus 2 is squared."},{"Start":"06:15.935 ","End":"06:23.720","Text":"This gives us x minus 2 and x minus 2 squared all consecutive powers up to 2."},{"Start":"06:23.720 ","End":"06:27.635","Text":"The quadratic factor is also squared,"},{"Start":"06:27.635 ","End":"06:31.310","Text":"and it contributes x squared plus 4 to the power"},{"Start":"06:31.310 ","End":"06:35.690","Text":"of 1 and x squared plus 4 to the power of 2."},{"Start":"06:35.690 ","End":"06:38.480","Text":"We just stop here because it\u0027s 2 here,"},{"Start":"06:38.480 ","End":"06:40.310","Text":"but whatever this was, if this was cubed,"},{"Start":"06:40.310 ","End":"06:42.710","Text":"we\u0027d have another factor and so on."},{"Start":"06:42.710 ","End":"06:45.350","Text":"For each linear factor,"},{"Start":"06:45.350 ","End":"06:47.855","Text":"we just put a constant on the numerator,"},{"Start":"06:47.855 ","End":"06:49.849","Text":"and for each quadratic factor,"},{"Start":"06:49.849 ","End":"06:53.150","Text":"we put a linear expression on the numerator."},{"Start":"06:53.150 ","End":"06:55.265","Text":"How do we continue from here?"},{"Start":"06:55.265 ","End":"06:58.440","Text":"Well, this brings us to step 3."}],"ID":4496},{"Watched":false,"Name":"General Case - Third Step","Duration":"8m 1s","ChapterTopicVideoID":4487,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"Now the third and final step."},{"Start":"00:03.480 ","End":"00:08.370","Text":"Well, I\u0027m presuming that you already know how to find the constants in this."},{"Start":"00:08.370 ","End":"00:10.860","Text":"We\u0027ve already learned how to do this and there\u0027s plenty of"},{"Start":"00:10.860 ","End":"00:14.625","Text":"examples in the exercises following the theoretical section."},{"Start":"00:14.625 ","End":"00:17.684","Text":"In fact, why don\u0027t we just start with one complete example."},{"Start":"00:17.684 ","End":"00:21.750","Text":"But I\u0027d like to emphasize that at this point, after we\u0027ve decomposed,"},{"Start":"00:21.750 ","End":"00:28.305","Text":"all we have left are immediate integrals and things of the form the integral"},{"Start":"00:28.305 ","End":"00:37.140","Text":"of mx plus n over x squared plus bx plus c dx,"},{"Start":"00:37.140 ","End":"00:39.080","Text":"or the possibly could be a constant here,"},{"Start":"00:39.080 ","End":"00:41.275","Text":"but we\u0027ve learned how to get rid of that constant."},{"Start":"00:41.275 ","End":"00:45.725","Text":"This is essentially the only thing we have left us immediate integrals."},{"Start":"00:45.725 ","End":"00:48.605","Text":"Let\u0027s just go on to that complete example."},{"Start":"00:48.605 ","End":"00:50.570","Text":"Here\u0027s our example, and we\u0027ll only do"},{"Start":"00:50.570 ","End":"00:53.450","Text":"one example because it\u0027s a lot of work and you have lots"},{"Start":"00:53.450 ","End":"00:58.280","Text":"of sample exercises following this tutorial section and please do them."},{"Start":"00:58.280 ","End":"00:59.510","Text":"You need a lot of practice."},{"Start":"00:59.510 ","End":"01:02.630","Text":"We\u0027re still talking about the degree in the numerator."},{"Start":"01:02.630 ","End":"01:07.455","Text":"In this case, it\u0027s 2 being less than a degree in the denominator,"},{"Start":"01:07.455 ","End":"01:08.720","Text":"and this we have here,"},{"Start":"01:08.720 ","End":"01:11.480","Text":"the general case is hard to do."},{"Start":"01:11.480 ","End":"01:14.960","Text":"In fact, I don\u0027t know if there is a general way of solving it,"},{"Start":"01:14.960 ","End":"01:17.465","Text":"but if you get this question on an exam or something,"},{"Start":"01:17.465 ","End":"01:20.555","Text":"you can be sure that one trick or another will work."},{"Start":"01:20.555 ","End":"01:24.530","Text":"In this case, we can factorize it using groups."},{"Start":"01:24.530 ","End":"01:27.260","Text":"I\u0027ll show you what I mean, I\u0027ll do it at the side."},{"Start":"01:27.260 ","End":"01:33.030","Text":"We have x cubed plus 2x squared plus x plus 2."},{"Start":"01:33.030 ","End":"01:34.485","Text":"That\u0027s the denominator."},{"Start":"01:34.485 ","End":"01:36.920","Text":"If I take x squared out of the first two terms,"},{"Start":"01:36.920 ","End":"01:38.645","Text":"I\u0027ll get x squared."},{"Start":"01:38.645 ","End":"01:41.910","Text":"What I have left is x plus 2."},{"Start":"01:41.910 ","End":"01:44.520","Text":"Now I see there\u0027s an x plus 2 here."},{"Start":"01:44.520 ","End":"01:48.525","Text":"I write this as 1 times x plus 2."},{"Start":"01:48.525 ","End":"01:51.524","Text":"Now I have x plus 2 in both places,"},{"Start":"01:51.524 ","End":"01:54.015","Text":"so I can take that outside the brackets."},{"Start":"01:54.015 ","End":"02:00.305","Text":"What I\u0027m left with is x squared plus 1 times x plus 2,"},{"Start":"02:00.305 ","End":"02:02.510","Text":"and I can\u0027t factorize it any further."},{"Start":"02:02.510 ","End":"02:04.925","Text":"This one does not factorize."},{"Start":"02:04.925 ","End":"02:07.434","Text":"If it factorized and it would have roots,"},{"Start":"02:07.434 ","End":"02:12.955","Text":"and we would be able to solve x squared plus 1 equals 0, which we can\u0027t."},{"Start":"02:12.955 ","End":"02:14.920","Text":"This remains like this."},{"Start":"02:14.920 ","End":"02:16.520","Text":"That\u0027s step 1."},{"Start":"02:16.520 ","End":"02:18.290","Text":"Step 2, as you recall,"},{"Start":"02:18.290 ","End":"02:22.675","Text":"is to decompose into partial fractions."},{"Start":"02:22.675 ","End":"02:27.510","Text":"The way it works is that each of these is only to the power of 1."},{"Start":"02:27.510 ","End":"02:31.310","Text":"We put a constant over this plus a linear expression"},{"Start":"02:31.310 ","End":"02:35.090","Text":"over this and we get the x squared plus 1 here,"},{"Start":"02:35.090 ","End":"02:43.475","Text":"came from here, that\u0027s the irreducible quadratic and this x plus 2 is our linear factor,"},{"Start":"02:43.475 ","End":"02:45.605","Text":"and it came from here."},{"Start":"02:45.605 ","End":"02:48.450","Text":"Note that over the linear factor we just put a constant,"},{"Start":"02:48.450 ","End":"02:50.509","Text":"and over the irreducible quadratic,"},{"Start":"02:50.509 ","End":"02:53.430","Text":"we put a linear like Ax plus b."},{"Start":"02:53.430 ","End":"02:55.320","Text":"Let\u0027s find A, B,"},{"Start":"02:55.320 ","End":"02:59.490","Text":"and C. What we\u0027ll do is we\u0027ll put a common denominator,"},{"Start":"02:59.490 ","End":"03:04.895","Text":"and the common denominator will be x squared plus 1 times x plus 2."},{"Start":"03:04.895 ","End":"03:06.485","Text":"That\u0027s what we got from here,"},{"Start":"03:06.485 ","End":"03:11.600","Text":"that this denominator is x squared plus 1, x plus 2."},{"Start":"03:11.600 ","End":"03:20.345","Text":"We get just 2x squared plus 2x plus 1 equals over here,"},{"Start":"03:20.345 ","End":"03:24.110","Text":"we have to multiply by x plus 2."},{"Start":"03:24.110 ","End":"03:25.670","Text":"That\u0027s the missing factor."},{"Start":"03:25.670 ","End":"03:30.345","Text":"It\u0027s Ax plus B, x plus 2."},{"Start":"03:30.345 ","End":"03:34.955","Text":"Similarly, C, the missing factor is x squared plus 1,"},{"Start":"03:34.955 ","End":"03:36.845","Text":"so this is what we get."},{"Start":"03:36.845 ","End":"03:39.080","Text":"Also remember although I write equals,"},{"Start":"03:39.080 ","End":"03:40.370","Text":"it\u0027s really an identity,"},{"Start":"03:40.370 ","End":"03:42.820","Text":"it means that for all x this is true."},{"Start":"03:42.820 ","End":"03:44.690","Text":"If it\u0027s true for all x,"},{"Start":"03:44.690 ","End":"03:48.185","Text":"then we can substitute any value of x we like."},{"Start":"03:48.185 ","End":"03:50.930","Text":"The first value I\u0027d like to substitute is"},{"Start":"03:50.930 ","End":"03:55.220","Text":"minus 2 because that will make this 0 and that\u0027s easy to compute."},{"Start":"03:55.220 ","End":"03:58.550","Text":"If I do substitute x equals minus 2,"},{"Start":"03:58.550 ","End":"04:00.950","Text":"what I get is, on the left-hand side,"},{"Start":"04:00.950 ","End":"04:04.220","Text":"twice minus 2 squared is 2 times 4 is 8,"},{"Start":"04:04.220 ","End":"04:07.125","Text":"minus 2 times minus 2 minus 4."},{"Start":"04:07.125 ","End":"04:09.700","Text":"Together, 8 minus 4 is 4 plus 1 is 5,"},{"Start":"04:09.700 ","End":"04:12.230","Text":"and we get 5 on the left-hand side."},{"Start":"04:12.230 ","End":"04:14.725","Text":"So 5 equals now,"},{"Start":"04:14.725 ","End":"04:18.510","Text":"if x is minus 2, this plus this is 0,"},{"Start":"04:18.510 ","End":"04:20.250","Text":"and times this is still 0,"},{"Start":"04:20.250 ","End":"04:25.530","Text":"so we\u0027re left with C times minus 2 squared,"},{"Start":"04:25.530 ","End":"04:28.290","Text":"is 4, plus 1 is 5."},{"Start":"04:28.290 ","End":"04:30.944","Text":"So if 5 is C times 5,"},{"Start":"04:30.944 ","End":"04:34.100","Text":"that gives us that C equals 1."},{"Start":"04:34.100 ","End":"04:38.150","Text":"There are no other x\u0027s that will make something 0."},{"Start":"04:38.150 ","End":"04:42.935","Text":"What I\u0027ll try is just x equals 0 because that might be easy to compute."},{"Start":"04:42.935 ","End":"04:47.810","Text":"If that is so then on the left-hand side we get 1,"},{"Start":"04:47.810 ","End":"04:50.170","Text":"which equals, if x is 0,"},{"Start":"04:50.170 ","End":"04:56.910","Text":"this is just B and this is just 2 plus, again, if x is 0,"},{"Start":"04:56.910 ","End":"05:00.750","Text":"this is 0 plus 1 is 1 times C plus C,"},{"Start":"05:00.750 ","End":"05:02.910","Text":"but C equals 1,"},{"Start":"05:02.910 ","End":"05:04.290","Text":"so if C equals 1,"},{"Start":"05:04.290 ","End":"05:06.795","Text":"I\u0027m left with 2B equals 0,"},{"Start":"05:06.795 ","End":"05:11.940","Text":"and I get that B equals 0."},{"Start":"05:11.940 ","End":"05:14.280","Text":"All we need now is A."},{"Start":"05:14.280 ","End":"05:16.070","Text":"Now, I need another equation."},{"Start":"05:16.070 ","End":"05:17.785","Text":"I can let x be anything."},{"Start":"05:17.785 ","End":"05:19.410","Text":"I\u0027ll try x equals 1,"},{"Start":"05:19.410 ","End":"05:22.610","Text":"nice small number, convenient to substitute."},{"Start":"05:22.610 ","End":"05:25.590","Text":"I get A plus B."}],"ID":4497},{"Watched":false,"Name":"Exercise 1","Duration":"2m 42s","ChapterTopicVideoID":4440,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"In this exercise, we have to find the integral of 1 over x^2 minus 4."},{"Start":"00:04.830 ","End":"00:06.420","Text":"Now, look at the denominator,"},{"Start":"00:06.420 ","End":"00:07.600","Text":"x^2 minus 4."},{"Start":"00:07.600 ","End":"00:08.865","Text":"That should ring a bell."},{"Start":"00:08.865 ","End":"00:11.430","Text":"It\u0027s one of the shortcut multiplication formulas,"},{"Start":"00:11.430 ","End":"00:13.065","Text":"the difference of squares,"},{"Start":"00:13.065 ","End":"00:18.060","Text":"x^2 minus 4 is equal to x minus 2, x plus 2."},{"Start":"00:18.060 ","End":"00:19.890","Text":"We can write this like this."},{"Start":"00:19.890 ","End":"00:23.175","Text":"Now, we see these factors decomposed like this,"},{"Start":"00:23.175 ","End":"00:25.425","Text":"we think of partial fractions."},{"Start":"00:25.425 ","End":"00:28.500","Text":"Best way is to imagine that someone got to"},{"Start":"00:28.500 ","End":"00:33.480","Text":"this expression by combining 2 separate expressions of this form,"},{"Start":"00:33.480 ","End":"00:35.325","Text":"something over x minus 2"},{"Start":"00:35.325 ","End":"00:37.095","Text":"plus something over x plus 2,"},{"Start":"00:37.095 ","End":"00:38.715","Text":"and he combine this to get this"},{"Start":"00:38.715 ","End":"00:41.045","Text":"and we\u0027re doing detective work to find the opposite,"},{"Start":"00:41.045 ","End":"00:42.965","Text":"we want to know what A and B are."},{"Start":"00:42.965 ","End":"00:45.770","Text":"What we\u0027ll do is put them over a common denominator,"},{"Start":"00:45.770 ","End":"00:47.390","Text":"x minus 2, x plus 2,"},{"Start":"00:47.390 ","End":"00:49.250","Text":"and then we\u0027ll just throw out the denominator."},{"Start":"00:49.250 ","End":"00:54.110","Text":"In short, what we get is that we have the 1 here."},{"Start":"00:54.110 ","End":"00:56.240","Text":"The A was missing x plus 2"},{"Start":"00:56.240 ","End":"00:58.715","Text":"and the B was missing an x minus 2."},{"Start":"00:58.715 ","End":"01:00.559","Text":"We have this equation,"},{"Start":"01:00.559 ","End":"01:03.395","Text":"which is really an identity and not an equation,"},{"Start":"01:03.395 ","End":"01:05.195","Text":"meaning it\u0027s true for all x."},{"Start":"01:05.195 ","End":"01:08.555","Text":"If it\u0027s true for all x, then we can substitute whatever x we like."},{"Start":"01:08.555 ","End":"01:12.350","Text":"For example, if I substitute x equals 2"},{"Start":"01:12.350 ","End":"01:15.375","Text":"and the reason I\u0027m doing that is it will make something 0."},{"Start":"01:15.375 ","End":"01:16.870","Text":"If we make x equals 2,"},{"Start":"01:16.870 ","End":"01:20.930","Text":"we\u0027ll get 1 equals A times 4 plus B times 0."},{"Start":"01:20.930 ","End":"01:23.660","Text":"If we substitute x equals minus 2,"},{"Start":"01:23.660 ","End":"01:28.010","Text":"then we\u0027ll get 1 equals A times 0 plus B times minus 4."},{"Start":"01:28.010 ","End":"01:30.830","Text":"In other words, this one, if you look at it,"},{"Start":"01:30.830 ","End":"01:33.380","Text":"gives us B times 0 is 0,"},{"Start":"01:33.380 ","End":"01:35.640","Text":"that 4A is 1,"},{"Start":"01:35.640 ","End":"01:36.930","Text":"and if 4A is 1,"},{"Start":"01:36.930 ","End":"01:38.280","Text":"then A is 1/4."},{"Start":"01:38.280 ","End":"01:39.960","Text":"From this one, since this is 0,"},{"Start":"01:39.960 ","End":"01:43.170","Text":"we get B times minus 4 is 1,"},{"Start":"01:43.170 ","End":"01:46.000","Text":"so we get that B is minus 1/4."},{"Start":"01:46.000 ","End":"01:47.195","Text":"Now that we have this,"},{"Start":"01:47.195 ","End":"01:51.605","Text":"we can rewrite this as the integral of this."},{"Start":"01:51.605 ","End":"01:55.702","Text":"Of course, we can take the constants outside the integration sign,"},{"Start":"01:55.702 ","End":"01:59.695","Text":"the quarters can come out, and the plus becomes a plus,"},{"Start":"01:59.695 ","End":"02:03.495","Text":"so we get 1/4 times the integral of 1 over x minus 2"},{"Start":"02:03.495 ","End":"02:07.385","Text":"minus 1/4 times the integral of 1 over x plus 2."},{"Start":"02:07.385 ","End":"02:09.575","Text":"Now there\u0027s a formula that comes to our help here"},{"Start":"02:09.575 ","End":"02:11.725","Text":"that the integral of 1 over x plus something"},{"Start":"02:11.725 ","End":"02:14.170","Text":"is just the natural log of that something."},{"Start":"02:14.170 ","End":"02:18.300","Text":"In our case, what it becomes is we use it twice,"},{"Start":"02:18.300 ","End":"02:19.580","Text":"1/4 stays 1/4."},{"Start":"02:19.580 ","End":"02:22.174","Text":"This one becomes this from the formula,"},{"Start":"02:22.174 ","End":"02:24.560","Text":"and this becomes this from this formula,"},{"Start":"02:24.560 ","End":"02:27.740","Text":"and the constant of integration always at the end."},{"Start":"02:27.740 ","End":"02:30.830","Text":"Now, this will be good enough as an answer"},{"Start":"02:30.830 ","End":"02:34.910","Text":"but I could take the 1/4 outside the brackets, make it neater."},{"Start":"02:34.910 ","End":"02:39.280","Text":"When you have the logarithm of something minus the logarithm of something,"},{"Start":"02:39.280 ","End":"02:42.980","Text":"it\u0027s the logarithm of the quotient, and we\u0027re done."}],"ID":4449},{"Watched":false,"Name":"Exercise 2","Duration":"2m 13s","ChapterTopicVideoID":4441,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this exercise, we have to find the integral of 5 minus x"},{"Start":"00:04.020 ","End":"00:05.790","Text":"over x^2 plus 5x."},{"Start":"00:05.790 ","End":"00:09.000","Text":"Note that the denominator decomposes."},{"Start":"00:09.000 ","End":"00:13.225","Text":"That is, I can take x outside the brackets and what we get is"},{"Start":"00:13.225 ","End":"00:16.400","Text":"x, x plus 5 and the numerator as is."},{"Start":"00:16.400 ","End":"00:17.870","Text":"Now, when we see something like this,"},{"Start":"00:17.870 ","End":"00:21.155","Text":"we immediately think of partial fractions."},{"Start":"00:21.155 ","End":"00:23.840","Text":"It\u0027s as if someone started out with"},{"Start":"00:23.840 ","End":"00:27.440","Text":"an expression of something over x and something over x plus 5,"},{"Start":"00:27.440 ","End":"00:30.610","Text":"and then combined them to a common denominator to get this."},{"Start":"00:30.610 ","End":"00:34.205","Text":"Our job is to do the reverse and to find A and B."},{"Start":"00:34.205 ","End":"00:36.890","Text":"What we do is cross multiply here,"},{"Start":"00:36.890 ","End":"00:39.695","Text":"put over a common denominator, which is this."},{"Start":"00:39.695 ","End":"00:43.380","Text":"But then we throw away the denominator because they both have the same denominator."},{"Start":"00:43.380 ","End":"00:45.466","Text":"So we compare the numerators, we get this,"},{"Start":"00:45.466 ","End":"00:47.090","Text":"and it\u0027s not an equation,"},{"Start":"00:47.090 ","End":"00:51.470","Text":"it\u0027s an identity by which I mean that it\u0027s true for all x."},{"Start":"00:51.470 ","End":"00:53.060","Text":"If it\u0027s true for all x,"},{"Start":"00:53.060 ","End":"00:55.445","Text":"I can substitute any x I want,"},{"Start":"00:55.445 ","End":"00:59.360","Text":"and I usually choose x\u0027s that make something 0."},{"Start":"00:59.360 ","End":"01:02.240","Text":"For example, if I put x equals just 0,"},{"Start":"01:02.240 ","End":"01:04.970","Text":"then this becomes 0 and this is 0,"},{"Start":"01:04.970 ","End":"01:09.260","Text":"so 5 equals A times 5 plus B times 0."},{"Start":"01:09.260 ","End":"01:12.585","Text":"If I let x equals minus 5,"},{"Start":"01:12.585 ","End":"01:16.365","Text":"then we get minus 5 from 5 is 10,"},{"Start":"01:16.365 ","End":"01:20.085","Text":"A times 0, here, plus B times minus 5."},{"Start":"01:20.085 ","End":"01:23.646","Text":"From this one, I get A equals 1 because 5A is 5,"},{"Start":"01:23.646 ","End":"01:24.750","Text":"and from this one,"},{"Start":"01:24.750 ","End":"01:28.830","Text":"we get B times minus 5 is equal to 10,"},{"Start":"01:28.830 ","End":"01:30.500","Text":"so B is minus 2."},{"Start":"01:30.500 ","End":"01:32.130","Text":"Now that I have A and B,"},{"Start":"01:32.130 ","End":"01:36.841","Text":"I can put them here and so the integral becomes"},{"Start":"01:36.841 ","End":"01:42.170","Text":"1 over x, because A is 1, and minus 2 over x plus 5, because B is minus 2."},{"Start":"01:42.170 ","End":"01:46.965","Text":"I can split this into two and take any constants outside the brackets,"},{"Start":"01:46.965 ","End":"01:52.579","Text":"so I\u0027m left with the integral of this minus twice the integral of 1 over x plus 5."},{"Start":"01:52.579 ","End":"01:56.840","Text":"Now, we have a formula which tells us how to do both of these."},{"Start":"01:56.840 ","End":"01:58.200","Text":"When we have x plus a,"},{"Start":"01:58.200 ","End":"01:59.220","Text":"here, a is 5,"},{"Start":"01:59.220 ","End":"02:00.600","Text":"here, a is 0."},{"Start":"02:00.600 ","End":"02:02.165","Text":"If we apply this formula,"},{"Start":"02:02.165 ","End":"02:07.985","Text":"then we get natural log of x minus twice natural log of x plus 5."},{"Start":"02:07.985 ","End":"02:12.167","Text":"These are on absolute value and we have a single constant at the end."},{"Start":"02:12.167 ","End":"02:14.340","Text":"We\u0027re done."}],"ID":4450},{"Watched":false,"Name":"Exercise 3","Duration":"2m 36s","ChapterTopicVideoID":4442,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.150","Text":"In this exercise, we want to find the integral of x over x^2 plus 5x plus 6."},{"Start":"00:06.150 ","End":"00:09.195","Text":"We\u0027d like to factorize the denominator if possible."},{"Start":"00:09.195 ","End":"00:14.022","Text":"If it factorizes, then we can use the formula that"},{"Start":"00:14.022 ","End":"00:19.455","Text":"x^2 plus bx plus c is x minus 1 of the roots times x minus the other root,"},{"Start":"00:19.455 ","End":"00:20.770","Text":"assuming we have roots."},{"Start":"00:20.770 ","End":"00:24.430","Text":"The way we find roots is by comparing this expression to 0,"},{"Start":"00:24.430 ","End":"00:25.560","Text":"or in this case,"},{"Start":"00:25.560 ","End":"00:28.995","Text":"we have to compare the denominator here to 0."},{"Start":"00:28.995 ","End":"00:30.499","Text":"If we solve the quadratic equation,"},{"Start":"00:30.499 ","End":"00:32.205","Text":"it has 2 solutions, indeed."},{"Start":"00:32.205 ","End":"00:36.375","Text":"It has minus 3 and minus 2 as solutions."},{"Start":"00:36.375 ","End":"00:40.525","Text":"That makes our x_1 and x_2 to be minus 3 and minus 2."},{"Start":"00:40.525 ","End":"00:42.635","Text":"First of all, we can write this like this,"},{"Start":"00:42.635 ","End":"00:46.310","Text":"and then we can write this with the denominator factorized."},{"Start":"00:46.310 ","End":"00:50.000","Text":"Now, this algebraic expression was created"},{"Start":"00:50.000 ","End":"00:53.630","Text":"by someone by combining 2 expressions like this,"},{"Start":"00:53.630 ","End":"00:56.230","Text":"putting them under a common denominator and getting this."},{"Start":"00:56.230 ","End":"00:59.710","Text":"We have to do the reverse process and guess what A and B are."},{"Start":"00:59.710 ","End":"01:03.530","Text":"What we\u0027ll do is we\u0027ll put them over a common denominator,"},{"Start":"01:03.530 ","End":"01:05.630","Text":"which is this x plus 3, x plus 2,"},{"Start":"01:05.630 ","End":"01:07.565","Text":"and then compare the numerator."},{"Start":"01:07.565 ","End":"01:09.314","Text":"What we get is,"},{"Start":"01:09.314 ","End":"01:11.590","Text":"if we multiply by x plus 3, x plus 2,"},{"Start":"01:11.590 ","End":"01:12.980","Text":"x just stays x."},{"Start":"01:12.980 ","End":"01:15.710","Text":"A is missing a factor of x plus 2"},{"Start":"01:15.710 ","End":"01:17.930","Text":"and B is missing the x plus 3."},{"Start":"01:17.930 ","End":"01:19.295","Text":"This is what we get."},{"Start":"01:19.295 ","End":"01:21.350","Text":"This, of course, is not really an equation,"},{"Start":"01:21.350 ","End":"01:24.650","Text":"it\u0027s an identity which means that it works for every x."},{"Start":"01:24.650 ","End":"01:27.875","Text":"For example, if I put x equals minus 3,"},{"Start":"01:27.875 ","End":"01:29.000","Text":"and I did that deliberately,"},{"Start":"01:29.000 ","End":"01:30.350","Text":"so I\u0027ll get a 0 here,"},{"Start":"01:30.350 ","End":"01:35.220","Text":"then we get minus 3 is A times minus 1 plus 0B."},{"Start":"01:35.220 ","End":"01:41.149","Text":"Also, if we put x equals minus 2 to make the other thing 0,"},{"Start":"01:41.149 ","End":"01:45.590","Text":"we\u0027d get minus 2 is A times 0 plus B times 1."},{"Start":"01:45.590 ","End":"01:53.090","Text":"From the first, what we get is that A is 3 because we get that minus 3 is minus A."},{"Start":"01:53.090 ","End":"01:56.390","Text":"From here, we get minus 2 equals B,"},{"Start":"01:56.390 ","End":"01:59.360","Text":"so in other words, B is minus 2."},{"Start":"01:59.360 ","End":"02:00.560","Text":"Now that we have these,"},{"Start":"02:00.560 ","End":"02:02.255","Text":"we can put them in here and here,"},{"Start":"02:02.255 ","End":"02:04.220","Text":"and we get the integral of this,"},{"Start":"02:04.220 ","End":"02:05.880","Text":"which is equal to this."},{"Start":"02:05.880 ","End":"02:08.270","Text":"With integration, we can split it up into"},{"Start":"02:08.270 ","End":"02:11.410","Text":"the pluses and take constants outside the denominator."},{"Start":"02:11.410 ","End":"02:17.660","Text":"We get this expression, 3 times this integral minus 2 times this integral."},{"Start":"02:17.660 ","End":"02:24.365","Text":"We have a formula to come to our aid where we\u0027re going to use a as 3 and then as 2."},{"Start":"02:24.365 ","End":"02:28.580","Text":"Just apply this, not forgetting the 3 and the minus 2,"},{"Start":"02:28.580 ","End":"02:31.460","Text":"so we get 3 times natural log of x plus 3"},{"Start":"02:31.460 ","End":"02:34.808","Text":"minus twice the log of x plus 2,"},{"Start":"02:34.808 ","End":"02:37.380","Text":"and we are done."}],"ID":4451},{"Watched":false,"Name":"Exercise 4","Duration":"3m 25s","ChapterTopicVideoID":4443,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"In this exercise, we have to compute the integral of"},{"Start":"00:02.880 ","End":"00:06.570","Text":"8x minus 1 over 2x squared minus 3x minus 2."},{"Start":"00:06.570 ","End":"00:12.420","Text":"Now, the thing to notice about this exercise is that the coefficient of x squared in"},{"Start":"00:12.420 ","End":"00:15.390","Text":"the denominator is not 1 and we\u0027re used"},{"Start":"00:15.390 ","End":"00:18.405","Text":"to dealing with the coefficient 1 in the denominator."},{"Start":"00:18.405 ","End":"00:19.874","Text":"But as you recall,"},{"Start":"00:19.874 ","End":"00:22.410","Text":"this is not a problem because we can take 2"},{"Start":"00:22.410 ","End":"00:25.905","Text":"outside the brackets and divide each term by 2,"},{"Start":"00:25.905 ","End":"00:30.900","Text":"leaves us with this, and then we can take 2 outside of the integral sign altogether."},{"Start":"00:30.900 ","End":"00:33.450","Text":"We have a half and then we have our usual case where"},{"Start":"00:33.450 ","End":"00:36.315","Text":"it\u0027s just x squared here in the denominator."},{"Start":"00:36.315 ","End":"00:37.830","Text":"We\u0027d like to factorize."},{"Start":"00:37.830 ","End":"00:43.440","Text":"Just like here, x squared plus bx plus c can factorize into x minus x1, x minus x2,"},{"Start":"00:43.440 ","End":"00:49.835","Text":"provided that x1 and x2 are the roots of the quadratic equation where this equals 0."},{"Start":"00:49.835 ","End":"00:51.440","Text":"Now it may or may not have roots,"},{"Start":"00:51.440 ","End":"00:55.175","Text":"but we\u0027re assuming that because they gave us this exercise that it will."},{"Start":"00:55.175 ","End":"00:59.210","Text":"What we do is we let this denominator equals 0 and solve it,"},{"Start":"00:59.210 ","End":"01:03.375","Text":"and it turns out that the roots are minus 0.5 and 2."},{"Start":"01:03.375 ","End":"01:04.460","Text":"Now that we have the roots,"},{"Start":"01:04.460 ","End":"01:08.990","Text":"we can substitute and we can get the factorization of this what was"},{"Start":"01:08.990 ","End":"01:16.085","Text":"the denominator as to x minus 0.5 which makes it plus and x minus 2."},{"Start":"01:16.085 ","End":"01:19.730","Text":"Now I can write the integral as above,"},{"Start":"01:19.730 ","End":"01:23.130","Text":"but the denominator has been factorized into this."},{"Start":"01:23.130 ","End":"01:26.705","Text":"Now it\u0027s a situation of partial fractions."},{"Start":"01:26.705 ","End":"01:31.580","Text":"We imagine or assume that someone got to this expression by"},{"Start":"01:31.580 ","End":"01:36.890","Text":"combining 2 separate expressions with these as denominators,"},{"Start":"01:36.890 ","End":"01:40.250","Text":"and somehow he put a common denominator and ended up with this,"},{"Start":"01:40.250 ","End":"01:43.360","Text":"and we\u0027re doing detective work to get back to A and B."},{"Start":"01:43.360 ","End":"01:46.340","Text":"The way we do this is we also put a common denominator,"},{"Start":"01:46.340 ","End":"01:48.830","Text":"but then dispense with the denominator."},{"Start":"01:48.830 ","End":"01:52.850","Text":"What we get is after we multiply by the common denominator,"},{"Start":"01:52.850 ","End":"01:54.800","Text":"just 8x minus 1 here."},{"Start":"01:54.800 ","End":"01:58.925","Text":"Here we had to multiply by x minus 2 first that was missing,"},{"Start":"01:58.925 ","End":"02:02.810","Text":"and here we had to multiply by the x plus 0.5."},{"Start":"02:02.810 ","End":"02:06.530","Text":"This here is an identity more than an equation,"},{"Start":"02:06.530 ","End":"02:10.160","Text":"which means that if we put any value of x and it will be true."},{"Start":"02:10.160 ","End":"02:12.305","Text":"We can use this to find A and B."},{"Start":"02:12.305 ","End":"02:15.530","Text":"If I substitute x equals minus 0.5,"},{"Start":"02:15.530 ","End":"02:16.550","Text":"and why did I do that?"},{"Start":"02:16.550 ","End":"02:19.020","Text":"In order to make this bracket 0."},{"Start":"02:19.020 ","End":"02:20.325","Text":"Here, the 0."},{"Start":"02:20.325 ","End":"02:21.710","Text":"If we do the computation,"},{"Start":"02:21.710 ","End":"02:23.300","Text":"this is what we get."},{"Start":"02:23.300 ","End":"02:29.690","Text":"Then this is 0, so A is just minus 5 over minus 2.5, which is 2."},{"Start":"02:29.690 ","End":"02:33.629","Text":"This time we\u0027ll substitute plus 2 to make this 0,"},{"Start":"02:33.629 ","End":"02:36.015","Text":"so we get this expression."},{"Start":"02:36.015 ","End":"02:41.280","Text":"This is 0, so B is 15 over 2.5 and 15 over 2.5 is 6."},{"Start":"02:41.280 ","End":"02:42.870","Text":"Now we have A and B."},{"Start":"02:42.870 ","End":"02:44.460","Text":"We can put them in here."},{"Start":"02:44.460 ","End":"02:50.145","Text":"We can now rewrite the 8x minus 1 over this thing as 1.5 with 1.5 here."},{"Start":"02:50.145 ","End":"02:55.300","Text":"This integral becomes this because we\u0027ve just taken this with A and B as 2 and 6,"},{"Start":"02:55.300 ","End":"02:59.535","Text":"and then we just throw the 1.5 back in."},{"Start":"02:59.535 ","End":"03:01.620","Text":"For the 2, we get 1, for the 6,"},{"Start":"03:01.620 ","End":"03:04.655","Text":"we get 3 and use this formula,"},{"Start":"03:04.655 ","End":"03:06.500","Text":"which I put in each exercise,"},{"Start":"03:06.500 ","End":"03:10.460","Text":"is that 1 over x plus a integral is this."},{"Start":"03:10.460 ","End":"03:13.039","Text":"I use it twice, once with a as 0.5,"},{"Start":"03:13.039 ","End":"03:15.035","Text":"1 with a as minus 2."},{"Start":"03:15.035 ","End":"03:20.495","Text":"I get finally that the answer is the natural log of x plus 0.5"},{"Start":"03:20.495 ","End":"03:26.850","Text":"plus 3 times the log of x minus 2 plus a constant."}],"ID":4452},{"Watched":false,"Name":"Exercise 5","Duration":"46s","ChapterTopicVideoID":4444,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.905","Text":"Here we have to compute the integral of 1 over x minus 4 squared."},{"Start":"00:04.905 ","End":"00:07.503","Text":"Now, the denominator is already factorized,"},{"Start":"00:07.503 ","End":"00:10.721","Text":"so what we\u0027re going to do is use the formula"},{"Start":"00:10.721 ","End":"00:14.430","Text":"that if we have x plus a constant to the power of n,"},{"Start":"00:14.430 ","End":"00:21.140","Text":"then its integral is the same thing to the power of n plus 1 divided by n plus 1."},{"Start":"00:21.140 ","End":"00:24.807","Text":"In our case, we can write this as to the power of minus 2,"},{"Start":"00:24.807 ","End":"00:27.140","Text":"instead of 2 on the denominator,"},{"Start":"00:27.140 ","End":"00:29.255","Text":"so n is minus 2."},{"Start":"00:29.255 ","End":"00:33.170","Text":"If we use the formula minus 2 plus 1 is just minus 1,"},{"Start":"00:33.170 ","End":"00:34.910","Text":"so it\u0027s x minus 4 to the minus 1"},{"Start":"00:34.910 ","End":"00:36.775","Text":"and we divide by the minus 1."},{"Start":"00:36.775 ","End":"00:40.115","Text":"At the end, we can write it as minus 1,"},{"Start":"00:40.115 ","End":"00:43.520","Text":"and we can put this back in the denominator like it was originally,"},{"Start":"00:43.520 ","End":"00:44.923","Text":"over x minus 4."},{"Start":"00:44.923 ","End":"00:47.070","Text":"That\u0027s it."}],"ID":4453},{"Watched":false,"Name":"Exercise 6","Duration":"2m 10s","ChapterTopicVideoID":4445,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.819","Text":"Here we have to find the integral of x plus 4 over x minus 1 squared."},{"Start":"00:05.819 ","End":"00:10.185","Text":"What we\u0027ll need to do is decompose into partial fractions."},{"Start":"00:10.185 ","End":"00:13.110","Text":"Because we have this linear factor squared,"},{"Start":"00:13.110 ","End":"00:18.000","Text":"then what we\u0027ll get is all the powers of x minus 1 up to 2."},{"Start":"00:18.000 ","End":"00:20.985","Text":"We have an x minus 1 and an x minus 1 squared,"},{"Start":"00:20.985 ","End":"00:23.145","Text":"each 1 with a constant over the top."},{"Start":"00:23.145 ","End":"00:28.445","Text":"It\u0027s as if someone started with this and put a common denominator and ended up with this."},{"Start":"00:28.445 ","End":"00:30.799","Text":"We\u0027ll put both sides over a common denominator,"},{"Start":"00:30.799 ","End":"00:33.265","Text":"which is x minus 1 squared."},{"Start":"00:33.265 ","End":"00:34.980","Text":"What we\u0027ll get is here,"},{"Start":"00:34.980 ","End":"00:37.420","Text":"just the x plus 4 and here the b,"},{"Start":"00:37.420 ","End":"00:39.485","Text":"they already are over the denominator."},{"Start":"00:39.485 ","End":"00:42.260","Text":"Here we\u0027re missing an x minus 1, which is here."},{"Start":"00:42.260 ","End":"00:44.030","Text":"This is the equation we get,"},{"Start":"00:44.030 ","End":"00:46.100","Text":"which is really more of an identity."},{"Start":"00:46.100 ","End":"00:48.660","Text":"Any value of x will work,"},{"Start":"00:48.660 ","End":"00:50.550","Text":"so let\u0027s try x equals 1,"},{"Start":"00:50.550 ","End":"00:52.155","Text":"that will make this disappear."},{"Start":"00:52.155 ","End":"00:56.915","Text":"Then we get that 5 equals a times 0 plus b."},{"Start":"00:56.915 ","End":"01:00.035","Text":"That gives us that b is equal to 5."},{"Start":"01:00.035 ","End":"01:02.855","Text":"If we substitute any other value,"},{"Start":"01:02.855 ","End":"01:05.075","Text":"let\u0027s say x equals 2."},{"Start":"01:05.075 ","End":"01:08.105","Text":"Then here we get 2 minus 1 is 1,"},{"Start":"01:08.105 ","End":"01:09.755","Text":"2 plus 4 is 6,"},{"Start":"01:09.755 ","End":"01:13.790","Text":"and we get 6 equals a times 1 plus b."},{"Start":"01:13.790 ","End":"01:16.625","Text":"But we already know that b is 5,"},{"Start":"01:16.625 ","End":"01:20.030","Text":"so 6 is a times 1 plus 5,"},{"Start":"01:20.030 ","End":"01:21.455","Text":"so a is equal to 1."},{"Start":"01:21.455 ","End":"01:23.045","Text":"Now we have b and a."},{"Start":"01:23.045 ","End":"01:29.985","Text":"We can plug those in here and here and get an alternative expression for this decomposed."},{"Start":"01:29.985 ","End":"01:36.425","Text":"We get the integral of 1 over x minus 1 plus 5 over x minus 1 squared."},{"Start":"01:36.425 ","End":"01:40.760","Text":"This we split into 2 bits and take the constant out."},{"Start":"01:40.760 ","End":"01:47.450","Text":"Then we have, I\u0027ll just rewrite it with the negative exponents so we can use the formula."},{"Start":"01:47.450 ","End":"01:51.960","Text":"Here we have the natural log because it\u0027s 1 over x plus a,"},{"Start":"01:51.960 ","End":"01:56.060","Text":"and here we have the x plus a to the power of n. We raise"},{"Start":"01:56.060 ","End":"02:01.350","Text":"the power by 1 from minus 2 to minus 1 and divide by the new power."},{"Start":"02:01.350 ","End":"02:05.270","Text":"Finally, we can just rewrite it without negative exponents."},{"Start":"02:05.270 ","End":"02:07.010","Text":"We put this into the denominator,"},{"Start":"02:07.010 ","End":"02:08.615","Text":"take the minus out front,"},{"Start":"02:08.615 ","End":"02:11.760","Text":"and this is the answer."}],"ID":4454},{"Watched":false,"Name":"Exercise 7","Duration":"2m 49s","ChapterTopicVideoID":4446,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"Here, we have the integral of 6 minus x"},{"Start":"00:03.150 ","End":"00:06.045","Text":"over x squared plus 8x plus 16."},{"Start":"00:06.045 ","End":"00:07.920","Text":"As usual, we begin by trying"},{"Start":"00:07.920 ","End":"00:09.960","Text":"to factorize the denominator,"},{"Start":"00:09.960 ","End":"00:12.660","Text":"and 1 way to factorize the denominator"},{"Start":"00:12.660 ","End":"00:15.120","Text":"is just to set it equal to 0"},{"Start":"00:15.120 ","End":"00:16.080","Text":"and find the roots."},{"Start":"00:16.080 ","End":"00:19.050","Text":"Well, if you do this using the formula,"},{"Start":"00:19.050 ","End":"00:21.030","Text":"you get x equals minus 4,"},{"Start":"00:21.030 ","End":"00:22.170","Text":"and that\u0027s the only 1."},{"Start":"00:22.170 ","End":"00:24.030","Text":"Sometimes we say it\u0027s a double root"},{"Start":"00:24.030 ","End":"00:25.560","Text":"minus 4 and minus 4,"},{"Start":"00:25.560 ","End":"00:26.925","Text":"2 roots, but they are the same."},{"Start":"00:26.925 ","End":"00:28.590","Text":"Then the event when this happens,"},{"Start":"00:28.590 ","End":"00:30.000","Text":"then we write the denominator"},{"Start":"00:30.000 ","End":"00:32.439","Text":"as x minus the root all squared."},{"Start":"00:32.439 ","End":"00:34.110","Text":"We have the 6 minus x here"},{"Start":"00:34.110 ","End":"00:36.030","Text":"and x minus, minus 4 all squared,"},{"Start":"00:36.030 ","End":"00:38.130","Text":"but minus, minus is plus."},{"Start":"00:38.130 ","End":"00:39.920","Text":"This is what we get when"},{"Start":"00:39.920 ","End":"00:41.390","Text":"we factorize the denominator."},{"Start":"00:41.390 ","End":"00:43.340","Text":"We only have a linear factor squared,"},{"Start":"00:43.340 ","End":"00:45.300","Text":"and we decompose into partial fractions,"},{"Start":"00:45.300 ","End":"00:47.120","Text":"we get a representative for each"},{"Start":"00:47.120 ","End":"00:50.150","Text":"exponent from 1-2 in this case."},{"Start":"00:50.150 ","End":"00:53.450","Text":"What we get is a over x plus 4^1"},{"Start":"00:53.450 ","End":"00:55.985","Text":"plus b over x plus 4^2."},{"Start":"00:55.985 ","End":"00:58.160","Text":"You have to imagine that we got here"},{"Start":"00:58.160 ","End":"01:00.170","Text":"by starting off with something like this,"},{"Start":"01:00.170 ","End":"01:01.610","Text":"putting a common denominator"},{"Start":"01:01.610 ","End":"01:02.480","Text":"and getting to this,"},{"Start":"01:02.480 ","End":"01:04.080","Text":"and we\u0027re doing the reverse process."},{"Start":"01:04.080 ","End":"01:05.600","Text":"All we have to do is again"},{"Start":"01:05.600 ","End":"01:06.890","Text":"put a common denominator,"},{"Start":"01:06.890 ","End":"01:08.779","Text":"which will be x plus 4 squared."},{"Start":"01:08.779 ","End":"01:10.730","Text":"We put everything over this common denominator."},{"Start":"01:10.730 ","End":"01:13.535","Text":"This is as is and the B as is."},{"Start":"01:13.535 ","End":"01:15.770","Text":"But A is missing a factor of x plus 4,"},{"Start":"01:15.770 ","End":"01:16.625","Text":"and there it is."},{"Start":"01:16.625 ","End":"01:18.140","Text":"This is the equation we get."},{"Start":"01:18.140 ","End":"01:20.840","Text":"It\u0027s more really an identity which means"},{"Start":"01:20.840 ","End":"01:23.830","Text":"that every x that we put it will hold."},{"Start":"01:23.830 ","End":"01:25.790","Text":"Let\u0027s put x equals minus 4"},{"Start":"01:25.790 ","End":"01:27.965","Text":"to make this 0 first of all."},{"Start":"01:27.965 ","End":"01:29.690","Text":"Then we get on this side, 10"},{"Start":"01:29.690 ","End":"01:32.900","Text":"and on this side, A times 0 plus B,"},{"Start":"01:32.900 ","End":"01:35.900","Text":"and that leaves us with B equals 10."},{"Start":"01:35.900 ","End":"01:37.460","Text":"Then another value of x"},{"Start":"01:37.460 ","End":"01:38.360","Text":"that we could take,"},{"Start":"01:38.360 ","End":"01:41.405","Text":"just say minus 3, for example,"},{"Start":"01:41.405 ","End":"01:43.370","Text":"that would give me here 9,"},{"Start":"01:43.370 ","End":"01:44.780","Text":"and here 1."},{"Start":"01:44.780 ","End":"01:49.905","Text":"We would get that A plus B is 9."},{"Start":"01:49.905 ","End":"01:51.735","Text":"But since B is 10,"},{"Start":"01:51.735 ","End":"01:54.630","Text":"then A is minus 1."},{"Start":"01:54.630 ","End":"01:56.374","Text":"Now, that we have these constants,"},{"Start":"01:56.374 ","End":"01:59.419","Text":"we can plug them in here and here."},{"Start":"01:59.419 ","End":"02:01.790","Text":"Then we can rewrite our integral"},{"Start":"02:01.790 ","End":"02:03.980","Text":"as the integral of A,"},{"Start":"02:03.980 ","End":"02:06.200","Text":"which is minus 1 over x plus 4,"},{"Start":"02:06.200 ","End":"02:09.475","Text":"plus B, which is 10 over x plus 4 squared."},{"Start":"02:09.475 ","End":"02:11.690","Text":"We can split these constants"},{"Start":"02:11.690 ","End":"02:13.370","Text":"come out of the integral sign,"},{"Start":"02:13.370 ","End":"02:14.630","Text":"a plus becomes a plus,"},{"Start":"02:14.630 ","End":"02:16.310","Text":"so we have 2 separate integrals"},{"Start":"02:16.310 ","End":"02:17.420","Text":"with constants."},{"Start":"02:17.420 ","End":"02:20.320","Text":"I just rewrite this to the power of minus 2,"},{"Start":"02:20.320 ","End":"02:24.920","Text":"so we can use the formula for x plus a^n."},{"Start":"02:24.920 ","End":"02:26.840","Text":"In this case, we have a formula"},{"Start":"02:26.840 ","End":"02:29.035","Text":"which defines this as the logarithm."},{"Start":"02:29.035 ","End":"02:30.320","Text":"Here, we have the formula"},{"Start":"02:30.320 ","End":"02:31.940","Text":"where we raise the power by 1"},{"Start":"02:31.940 ","End":"02:33.830","Text":"and divide by that power."},{"Start":"02:33.830 ","End":"02:35.690","Text":"For minus 2, we get minus 1,"},{"Start":"02:35.690 ","End":"02:37.460","Text":"and we divide by minus 1."},{"Start":"02:37.460 ","End":"02:39.080","Text":"If we just simplify this,"},{"Start":"02:39.080 ","End":"02:40.700","Text":"we get the 10 here,"},{"Start":"02:40.700 ","End":"02:42.979","Text":"combined with the minus we put in front,"},{"Start":"02:42.979 ","End":"02:44.660","Text":"and the x plus 4 goes down"},{"Start":"02:44.660 ","End":"02:45.730","Text":"to the denominator,"},{"Start":"02:45.730 ","End":"02:46.970","Text":"and this is just slightly"},{"Start":"02:46.970 ","End":"02:48.500","Text":"neater to write it this way."},{"Start":"02:48.500 ","End":"02:50.430","Text":"Anyway, we\u0027re done."}],"ID":4455},{"Watched":false,"Name":"Exercise 8","Duration":"1m 25s","ChapterTopicVideoID":4447,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"Here we have the integral of 2x over x squared plus 5."},{"Start":"00:03.675 ","End":"00:09.165","Text":"Now take note that the derivative of the denominator is the numerator 2x,"},{"Start":"00:09.165 ","End":"00:12.480","Text":"and what we have is a formula to such cases that"},{"Start":"00:12.480 ","End":"00:15.810","Text":"when we have the derivative of a function over a function,"},{"Start":"00:15.810 ","End":"00:20.430","Text":"the integral of it is simply the natural log of the absolute value of the function."},{"Start":"00:20.430 ","End":"00:23.010","Text":"What we get in this case is just applying"},{"Start":"00:23.010 ","End":"00:26.635","Text":"the formula to natural log of the denominator plus the constant."},{"Start":"00:26.635 ","End":"00:28.760","Text":"If we look at the next example,"},{"Start":"00:28.760 ","End":"00:30.425","Text":"we\u0027ll also going to use this formula,"},{"Start":"00:30.425 ","End":"00:33.350","Text":"but it doesn\u0027t quite work here straight away"},{"Start":"00:33.350 ","End":"00:36.620","Text":"because the derivative of the denominator is 2x and not x,"},{"Start":"00:36.620 ","End":"00:37.940","Text":"so what are we going to do?"},{"Start":"00:37.940 ","End":"00:39.035","Text":"Well, the answer is simple."},{"Start":"00:39.035 ","End":"00:40.310","Text":"We just make a correction."},{"Start":"00:40.310 ","End":"00:41.630","Text":"We want 2x here,"},{"Start":"00:41.630 ","End":"00:42.830","Text":"so we put 2x here,"},{"Start":"00:42.830 ","End":"00:47.280","Text":"but of course we have to compensate by putting a 2 in the denominator also."},{"Start":"00:47.280 ","End":"00:51.500","Text":"Now we can integrate this using this formula and the 1/2 stays in front."},{"Start":"00:51.500 ","End":"00:57.170","Text":"Here we have another example where the derivative of the denominator is not 2x here,"},{"Start":"00:57.170 ","End":"01:00.065","Text":"is not even x here, it\u0027s something else, it\u0027s 7x."},{"Start":"01:00.065 ","End":"01:01.730","Text":"What we\u0027d like to do, first of all,"},{"Start":"01:01.730 ","End":"01:02.795","Text":"take the 7 out,"},{"Start":"01:02.795 ","End":"01:06.095","Text":"then we\u0027re back to the previous case and then we can play around with the 2,"},{"Start":"01:06.095 ","End":"01:09.020","Text":"so it\u0027s the formula to remind us."},{"Start":"01:09.020 ","End":"01:11.090","Text":"What we do is take the 7 out."},{"Start":"01:11.090 ","End":"01:13.880","Text":"Now we did use the previous trick of putting the 2"},{"Start":"01:13.880 ","End":"01:17.000","Text":"here and compensating by putting a 2 in the denominator,"},{"Start":"01:17.000 ","End":"01:18.680","Text":"so we haven\u0027t changed the exercise,"},{"Start":"01:18.680 ","End":"01:21.410","Text":"and now we have 7 over 2 and we use the formula."},{"Start":"01:21.410 ","End":"01:25.980","Text":"We get 7 over 2 natural log of the denominator, and that\u0027s it."}],"ID":4456},{"Watched":false,"Name":"Exercise 9","Duration":"1m 13s","ChapterTopicVideoID":4448,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"Here, I\u0027d like to just show you some formulas for use in the following exercises."},{"Start":"00:04.950 ","End":"00:09.780","Text":"There is an immediate integral that the integral of 1 over x^2 plus 1 is,"},{"Start":"00:09.780 ","End":"00:13.064","Text":"simply, the arctangent of x plus the constant."},{"Start":"00:13.064 ","End":"00:14.990","Text":"Someone forgot to write a dx here."},{"Start":"00:14.990 ","End":"00:18.600","Text":"This can be generalized to the following, that if,"},{"Start":"00:18.600 ","End":"00:20.190","Text":"instead of 1, we have c,"},{"Start":"00:20.190 ","End":"00:22.365","Text":"c has to be a positive constant."},{"Start":"00:22.365 ","End":"00:25.005","Text":"For example, we want to take its square root,"},{"Start":"00:25.005 ","End":"00:27.300","Text":"then this is equal to 1 over the square root of c,"},{"Start":"00:27.300 ","End":"00:29.753","Text":"the arctangent of x over the square root of c."},{"Start":"00:29.753 ","End":"00:31.680","Text":"Notice that if c is 1,"},{"Start":"00:31.680 ","End":"00:33.862","Text":"we get exactly what we wrote here"},{"Start":"00:33.862 ","End":"00:36.480","Text":"because x over square root of c is just x."},{"Start":"00:36.480 ","End":"00:38.985","Text":"This will be useful to us later."},{"Start":"00:38.985 ","End":"00:42.250","Text":"Just to show you an example where c is 4."},{"Start":"00:42.250 ","End":"00:47.285","Text":"The integral of 1 over x^2 plus 4 would be 1 over the square root of 4,"},{"Start":"00:47.285 ","End":"00:50.000","Text":"arctangent of x over the square root of 4,"},{"Start":"00:50.000 ","End":"00:52.105","Text":"square root of 4 is 2, of course."},{"Start":"00:52.105 ","End":"00:54.400","Text":"Here\u0027s another example."},{"Start":"00:54.400 ","End":"00:56.780","Text":"This time, it isn\u0027t even 1 on the top."},{"Start":"00:56.780 ","End":"00:59.305","Text":"It\u0027s 4 over x^2 plus 10."},{"Start":"00:59.305 ","End":"01:01.590","Text":"We take the 4 outside the integrals,"},{"Start":"01:01.590 ","End":"01:02.850","Text":"so the 4 stays."},{"Start":"01:02.850 ","End":"01:05.341","Text":"Then we have the integral of 1 over x^2 plus 10,"},{"Start":"01:05.341 ","End":"01:08.840","Text":"which from here, just gives us this with c is equal to 10,"},{"Start":"01:08.840 ","End":"01:10.505","Text":"and this is what we get."},{"Start":"01:10.505 ","End":"01:14.340","Text":"That was just some theory, and I\u0027m done."}],"ID":4457},{"Watched":false,"Name":"Exercise 10","Duration":"47s","ChapterTopicVideoID":4449,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.195","Text":"In this exercise, we have to compute the following integral."},{"Start":"00:03.195 ","End":"00:05.370","Text":"Let\u0027s split it up into 2,"},{"Start":"00:05.370 ","End":"00:08.280","Text":"the 2x separately, and the 1 separately."},{"Start":"00:08.280 ","End":"00:10.500","Text":"We\u0027ll get something like this,"},{"Start":"00:10.500 ","End":"00:12.765","Text":"2x over the denominator,"},{"Start":"00:12.765 ","End":"00:15.165","Text":"and 1 over that denominator."},{"Start":"00:15.165 ","End":"00:17.175","Text":"Now, each of these is different,"},{"Start":"00:17.175 ","End":"00:21.200","Text":"and each will need this formula for"},{"Start":"00:21.200 ","End":"00:25.459","Text":"this because here we have the derivative of the denominator and the numerator,"},{"Start":"00:25.459 ","End":"00:27.410","Text":"so it will come out as a logarithm."},{"Start":"00:27.410 ","End":"00:29.840","Text":"Whereas here, we have something of this form,"},{"Start":"00:29.840 ","End":"00:31.330","Text":"1 over x squared plus c."},{"Start":"00:31.330 ","End":"00:33.345","Text":"So we\u0027ll get something with an arctangent."},{"Start":"00:33.345 ","End":"00:37.475","Text":"Specifically, here we\u0027ll get the natural log of x squared plus 1,"},{"Start":"00:37.475 ","End":"00:39.785","Text":"and here c is equal to 1,"},{"Start":"00:39.785 ","End":"00:42.005","Text":"so square root of c is also 1."},{"Start":"00:42.005 ","End":"00:44.220","Text":"We just get the arctangent of x,"},{"Start":"00:44.220 ","End":"00:48.570","Text":"and of course, there\u0027s a single constant at the end, and that\u0027s it."}],"ID":4458},{"Watched":false,"Name":"Exercise 11","Duration":"1m ","ChapterTopicVideoID":4450,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this exercise, we want the integral of 4x plus 10 over x squared plus 9."},{"Start":"00:05.070 ","End":"00:07.275","Text":"Let\u0027s split it up into 2 pieces,"},{"Start":"00:07.275 ","End":"00:10.230","Text":"the 4x over this and 10 over this."},{"Start":"00:10.230 ","End":"00:12.945","Text":"But I\u0027m going to just skip a step instead of 4x,"},{"Start":"00:12.945 ","End":"00:15.300","Text":"I\u0027m going to write it as twice 2x."},{"Start":"00:15.300 ","End":"00:18.720","Text":"The reason I wrote it as twice 2x is that I could"},{"Start":"00:18.720 ","End":"00:22.485","Text":"get the denominator and its derivative on top,"},{"Start":"00:22.485 ","End":"00:24.900","Text":"so 4 is 2 times 2, so that\u0027s fine."},{"Start":"00:24.900 ","End":"00:26.295","Text":"The 10 comes out here."},{"Start":"00:26.295 ","End":"00:27.990","Text":"Now we need to separate formulas,"},{"Start":"00:27.990 ","End":"00:29.320","Text":"1 for each piece."},{"Start":"00:29.320 ","End":"00:32.825","Text":"This will be based on the paradigm of a function"},{"Start":"00:32.825 ","End":"00:36.635","Text":"and its derivative on top and that uses the natural logarithm."},{"Start":"00:36.635 ","End":"00:40.490","Text":"Here we have something of this form which uses arctangent."},{"Start":"00:40.490 ","End":"00:43.705","Text":"If we put c equals 9 here,"},{"Start":"00:43.705 ","End":"00:46.550","Text":"then what we\u0027ll get is from here,"},{"Start":"00:46.550 ","End":"00:48.260","Text":"we get the 2 from here,"},{"Start":"00:48.260 ","End":"00:50.900","Text":"the natural logarithm 10 here."},{"Start":"00:50.900 ","End":"00:53.330","Text":"If we just use the formula, this is what we get."},{"Start":"00:53.330 ","End":"00:55.955","Text":"Then we change the square root of 9 to 3."},{"Start":"00:55.955 ","End":"01:01.230","Text":"We have this expression with the arctangent just by substitution. That\u0027s it."}],"ID":4459},{"Watched":false,"Name":"Exercise 12","Duration":"1m 9s","ChapterTopicVideoID":4451,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.475","Text":"Here we have to compute the integral of 5x plus 6 over x squared plus 3."},{"Start":"00:05.475 ","End":"00:07.860","Text":"The denominator is irreducible."},{"Start":"00:07.860 ","End":"00:10.515","Text":"Let\u0027s split it up with this plus here."},{"Start":"00:10.515 ","End":"00:12.720","Text":"Let\u0027s also take constants out."},{"Start":"00:12.720 ","End":"00:19.530","Text":"What we get is 5 times integral of x over this plus 6 times integral of 1 over this."},{"Start":"00:19.530 ","End":"00:21.390","Text":"Now, each of these is different."},{"Start":"00:21.390 ","End":"00:25.110","Text":"This one, we want the numerator as the derivative of the denominator."},{"Start":"00:25.110 ","End":"00:27.435","Text":"If we want that, we need a 2 here."},{"Start":"00:27.435 ","End":"00:28.920","Text":"We put the 2 here,"},{"Start":"00:28.920 ","End":"00:31.215","Text":"but then we have to compensate with a 2 here."},{"Start":"00:31.215 ","End":"00:35.445","Text":"Now, this looks like the function derivative over a function,"},{"Start":"00:35.445 ","End":"00:41.090","Text":"that\u0027s this case, f prime over f. We\u0027ve got the formula for that natural logarithm."},{"Start":"00:41.090 ","End":"00:44.000","Text":"This 1 is one of those arc-tangent ones where we have on"},{"Start":"00:44.000 ","End":"00:48.960","Text":"the denominator x squared plus a constant which is 3."},{"Start":"00:48.960 ","End":"00:50.420","Text":"For the first bit,"},{"Start":"00:50.420 ","End":"00:56.000","Text":"we get 5 over 2 times the natural log of the x squared plus 3."},{"Start":"00:56.000 ","End":"00:59.000","Text":"We don\u0027t need absolute value because this thing is positive."},{"Start":"00:59.000 ","End":"01:02.765","Text":"On the second bit, we have the 6 from here and"},{"Start":"01:02.765 ","End":"01:07.455","Text":"1 over square root of c is 1 over square root of 3 here and here,"},{"Start":"01:07.455 ","End":"01:10.209","Text":"and finally, plus the constant."}],"ID":4460},{"Watched":false,"Name":"Exercise 13","Duration":"1m 49s","ChapterTopicVideoID":4452,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.190","Text":"Here we have to compute the integral of 1 over x^2 plus 2x plus 5."},{"Start":"00:05.190 ","End":"00:08.250","Text":"You can easily check that this is an irreducible quadratic,"},{"Start":"00:08.250 ","End":"00:10.995","Text":"it has no roots, and what we\u0027re going to do"},{"Start":"00:10.995 ","End":"00:14.670","Text":"is the trick of getting rid of the middle term,"},{"Start":"00:14.670 ","End":"00:18.210","Text":"the x term, by completing the square method."},{"Start":"00:18.210 ","End":"00:19.830","Text":"What we do is, first of all,"},{"Start":"00:19.830 ","End":"00:22.269","Text":"we remind ourselves of the formula we\u0027re going to use"},{"Start":"00:22.269 ","End":"00:24.405","Text":"that if we have x^2 plus bx,"},{"Start":"00:24.405 ","End":"00:27.420","Text":"we can write it as something squared minus"},{"Start":"00:27.420 ","End":"00:30.570","Text":"a number without the x term, and this is what it is."},{"Start":"00:30.570 ","End":"00:32.670","Text":"It\u0027s half this coefficient here,"},{"Start":"00:32.670 ","End":"00:34.300","Text":"x plus b over 2 squared."},{"Start":"00:34.300 ","End":"00:35.820","Text":"If you multiply this out,"},{"Start":"00:35.820 ","End":"00:39.030","Text":"you see that we have to subtract b over 2 squared to get it to equal."},{"Start":"00:39.030 ","End":"00:43.225","Text":"In our case, what we get is the x^2 plus 2x."},{"Start":"00:43.225 ","End":"00:46.550","Text":"These 2 terms will be equal to b over 2 is 1,"},{"Start":"00:46.550 ","End":"00:49.580","Text":"so it\u0027s x plus 1 squared minus 1 squared,"},{"Start":"00:49.580 ","End":"00:54.050","Text":"and that makes the whole denominator with the 5 equal to what\u0027s here,"},{"Start":"00:54.050 ","End":"00:57.590","Text":"plus the 5, and minus 1 plus 5 is 4."},{"Start":"00:57.590 ","End":"00:59.660","Text":"What we get here is this,"},{"Start":"00:59.660 ","End":"01:03.065","Text":"and this was just algebra so far to get from this to this."},{"Start":"01:03.065 ","End":"01:05.225","Text":"Now, when we have a case like this,"},{"Start":"01:05.225 ","End":"01:07.970","Text":"we use a substitution where we let this thing that\u0027s"},{"Start":"01:07.970 ","End":"01:09.250","Text":"squared be equal to t."},{"Start":"01:09.250 ","End":"01:11.810","Text":"Substitute t equals x plus 1"},{"Start":"01:11.810 ","End":"01:14.300","Text":"and then dt will equal dx."},{"Start":"01:14.300 ","End":"01:19.220","Text":"After the substitution, the integral becomes 1 over t squared plus 4"},{"Start":"01:19.220 ","End":"01:20.510","Text":"and dx is dt."},{"Start":"01:20.510 ","End":"01:24.576","Text":"Now, the formula that we need is this formula."},{"Start":"01:24.576 ","End":"01:25.885","Text":"Normally, it\u0027s with x,"},{"Start":"01:25.885 ","End":"01:28.290","Text":"now we write it with t because we have a t."},{"Start":"01:28.290 ","End":"01:31.220","Text":"It\u0027s this thing with the arctangent and the square root,"},{"Start":"01:31.220 ","End":"01:32.555","Text":"and in our case,"},{"Start":"01:32.555 ","End":"01:35.300","Text":"this gives with a is equaling 4,"},{"Start":"01:35.300 ","End":"01:37.490","Text":"so we get 1 over root 4"},{"Start":"01:37.490 ","End":"01:39.190","Text":"arctangent t over root 4,"},{"Start":"01:39.190 ","End":"01:41.970","Text":"and of course, root 4 is just 2."},{"Start":"01:41.970 ","End":"01:46.190","Text":"More than that, we can also replace the t by the x plus 1,"},{"Start":"01:46.190 ","End":"01:47.552","Text":"which originally it was,"},{"Start":"01:47.552 ","End":"01:50.910","Text":"and this gives us our answer. We are done."}],"ID":4461},{"Watched":false,"Name":"Exercise 14","Duration":"1m 31s","ChapterTopicVideoID":4453,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.700","Text":"Here we have to compute the integral of 1 over x^2 plus 4x plus 13."},{"Start":"00:05.700 ","End":"00:08.955","Text":"This on the denominator is an irreducible quadratic."},{"Start":"00:08.955 ","End":"00:10.020","Text":"You could try and solve it,"},{"Start":"00:10.020 ","End":"00:11.820","Text":"but you\u0027ll see that it has no solutions,"},{"Start":"00:11.820 ","End":"00:13.455","Text":"and so it\u0027s irreducible."},{"Start":"00:13.455 ","End":"00:16.305","Text":"We use the technique of completing the square"},{"Start":"00:16.305 ","End":"00:17.931","Text":"to try and get rid of the middle term"},{"Start":"00:17.931 ","End":"00:21.365","Text":"and get something with just something squared and a constant."},{"Start":"00:21.365 ","End":"00:23.420","Text":"The way we do this is, first of all,"},{"Start":"00:23.420 ","End":"00:27.064","Text":"recall the formula that x^2 plus bx."},{"Start":"00:27.064 ","End":"00:30.125","Text":"What we do is take x plus half this coefficient,"},{"Start":"00:30.125 ","End":"00:32.330","Text":"all squared, and it doesn\u0027t quite equal."},{"Start":"00:32.330 ","End":"00:35.180","Text":"We have to remove this bit to make the equality hold."},{"Start":"00:35.180 ","End":"00:37.220","Text":"Now, in our case, we take b as 4,"},{"Start":"00:37.220 ","End":"00:40.580","Text":"and the first 2 terms now become x plus 2,"},{"Start":"00:40.580 ","End":"00:41.870","Text":"all squared, minus 4,"},{"Start":"00:41.870 ","End":"00:44.380","Text":"simply by letting b equals 4 here."},{"Start":"00:44.380 ","End":"00:45.740","Text":"Then if we add the 13,"},{"Start":"00:45.740 ","End":"00:47.765","Text":"minus 4 plus 13,"},{"Start":"00:47.765 ","End":"00:49.700","Text":"that will give us 9 here."},{"Start":"00:49.700 ","End":"00:53.530","Text":"We get over here just by algebraic manipulation,"},{"Start":"00:53.530 ","End":"00:55.115","Text":"this is equal to this thing,"},{"Start":"00:55.115 ","End":"00:57.305","Text":"but now we don\u0027t have a middle term."},{"Start":"00:57.305 ","End":"01:00.590","Text":"Even more so when we substitute t equals x plus 2"},{"Start":"01:00.590 ","End":"01:02.405","Text":"and dt will equal dx,"},{"Start":"01:02.405 ","End":"01:06.925","Text":"then what we get is 1 over t^2 plus 9 and dx is dt."},{"Start":"01:06.925 ","End":"01:08.360","Text":"Now we have an expression,"},{"Start":"01:08.360 ","End":"01:10.385","Text":"a quadratic without the middle term."},{"Start":"01:10.385 ","End":"01:13.250","Text":"Then we recall the formula for this thing,"},{"Start":"01:13.250 ","End":"01:17.010","Text":"which is the arctangent and the square root."},{"Start":"01:18.340 ","End":"01:22.220","Text":"Since a is 9, we get 1 over root 9,"},{"Start":"01:22.220 ","End":"01:24.410","Text":"arctangent of t over root 9."},{"Start":"01:24.410 ","End":"01:26.465","Text":"Of course, root 9 is 3."},{"Start":"01:26.465 ","End":"01:30.650","Text":"Besides that, we also substitute back from t to x plus 2,"},{"Start":"01:30.650 ","End":"01:32.790","Text":"and that\u0027s the answer."}],"ID":4462},{"Watched":false,"Name":"Exercise 15","Duration":"1m 39s","ChapterTopicVideoID":4454,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.905","Text":"Here we have to solve the integral of 1 over x^2 plus x plus 1."},{"Start":"00:04.905 ","End":"00:08.055","Text":"Now, the denominator is an irreducible quadratic."},{"Start":"00:08.055 ","End":"00:09.390","Text":"You can try and solve it,"},{"Start":"00:09.390 ","End":"00:12.015","Text":"assign it to zero, you\u0027ll see it has no solution."},{"Start":"00:12.015 ","End":"00:15.330","Text":"When this is the case, we usually use the trick of completing"},{"Start":"00:15.330 ","End":"00:19.140","Text":"the square in order to get rid of the middle term in the quadratic."},{"Start":"00:19.140 ","End":"00:23.490","Text":"So remembering the formula that x^2 plus bx,"},{"Start":"00:23.490 ","End":"00:26.730","Text":"you start off with x plus half of this coefficient squared."},{"Start":"00:26.730 ","End":"00:29.880","Text":"Then to compensate because this is not quite equal,"},{"Start":"00:29.880 ","End":"00:31.700","Text":"we subtract b over 2 squared."},{"Start":"00:31.700 ","End":"00:33.290","Text":"If you expand it, you\u0027ll see this is true."},{"Start":"00:33.290 ","End":"00:34.865","Text":"We\u0027ve used this formula before."},{"Start":"00:34.865 ","End":"00:37.880","Text":"In our case, with the x^2 plus x, b is 1,"},{"Start":"00:37.880 ","End":"00:42.455","Text":"and so we get that x^2 plus x is x plus 1/2 squared."},{"Start":"00:42.455 ","End":"00:45.290","Text":"Then 1/2 squared is a 1/4, which we subtract."},{"Start":"00:45.290 ","End":"00:47.990","Text":"Then not forgetting that we had another 1 here,"},{"Start":"00:47.990 ","End":"00:53.000","Text":"what we get is that this denominator is equal to this minus 1/4 plus 1."},{"Start":"00:53.000 ","End":"00:54.710","Text":"This is 3/4."},{"Start":"00:54.710 ","End":"00:58.235","Text":"Then we just substitute the denominator using this,"},{"Start":"00:58.235 ","End":"01:00.095","Text":"so we get x plus 1/2 squared"},{"Start":"01:00.095 ","End":"01:02.245","Text":"and this bit is the 3/4."},{"Start":"01:02.245 ","End":"01:09.495","Text":"Now we make a substitution that this thing will be t and dt is, therefore, equal to dx."},{"Start":"01:09.495 ","End":"01:12.920","Text":"What we get is, really, quadratic without a middle term,"},{"Start":"01:12.920 ","End":"01:15.500","Text":"we get t squared plus 3/4."},{"Start":"01:15.500 ","End":"01:17.465","Text":"Again, remembering this formula,"},{"Start":"01:17.465 ","End":"01:19.010","Text":"that 1 over t^2 plus a,"},{"Start":"01:19.010 ","End":"01:20.075","Text":"which is what we have here,"},{"Start":"01:20.075 ","End":"01:23.040","Text":"is this expression with the arctangent and the square root."},{"Start":"01:23.040 ","End":"01:25.455","Text":"The a here is 3/4,"},{"Start":"01:25.455 ","End":"01:32.015","Text":"so that gives us 1 over square root of 3/4 arctangent of t over square root of 3/4."},{"Start":"01:32.015 ","End":"01:36.830","Text":"Finally, we substitute back from t to x plus 1/2"},{"Start":"01:36.830 ","End":"01:40.770","Text":"and that gives us the result. We\u0027re done."}],"ID":4463},{"Watched":false,"Name":"Exercise 16","Duration":"3m 2s","ChapterTopicVideoID":4455,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.535","Text":"Here we have to compute the integral of 3x minus 7 over x squared plus 2x plus 5."},{"Start":"00:05.535 ","End":"00:07.080","Text":"The denominator is what\u0027s important,"},{"Start":"00:07.080 ","End":"00:09.330","Text":"we want to decompose it, but it\u0027s irreducible."},{"Start":"00:09.330 ","End":"00:10.830","Text":"You can check there are no roots,"},{"Start":"00:10.830 ","End":"00:13.080","Text":"no solution, to this thing equals 0."},{"Start":"00:13.080 ","End":"00:15.360","Text":"What we do is the completing"},{"Start":"00:15.360 ","End":"00:18.735","Text":"the square method to try and get rid of the middle term of the quadratic."},{"Start":"00:18.735 ","End":"00:20.070","Text":"When I say completing the square,"},{"Start":"00:20.070 ","End":"00:23.960","Text":"I mean we use this formula that\u0027s x squared plus 2x,"},{"Start":"00:23.960 ","End":"00:25.515","Text":"in this case b is 2,"},{"Start":"00:25.515 ","End":"00:27.915","Text":"and we can break it up like this."},{"Start":"00:27.915 ","End":"00:31.565","Text":"What we get is that x squared plus 2x following the formula,"},{"Start":"00:31.565 ","End":"00:33.890","Text":"is x plus 1 all squared,"},{"Start":"00:33.890 ","End":"00:35.515","Text":"b is 2 over 2 is 1,"},{"Start":"00:35.515 ","End":"00:37.434","Text":"so 1 minus 1 squared."},{"Start":"00:37.434 ","End":"00:42.905","Text":"Here we have plus 5, so we have to add that 5 and that gives us this thing."},{"Start":"00:42.905 ","End":"00:44.595","Text":"Minus 1 plus 5 is 4,"},{"Start":"00:44.595 ","End":"00:50.230","Text":"so the integral now becomes 3x minus 7 over x plus 1 squared plus 4."},{"Start":"00:50.230 ","End":"00:53.075","Text":"So far with just an algebra, no integration yet."},{"Start":"00:53.075 ","End":"00:56.180","Text":"Now a substitution is in order that\u0027s the usual procedure."},{"Start":"00:56.180 ","End":"00:59.975","Text":"We let t equals x plus 1 and then dt is dx."},{"Start":"00:59.975 ","End":"01:02.255","Text":"Let\u0027s see what we get after the substitution."},{"Start":"01:02.255 ","End":"01:05.060","Text":"Well, the denominator is easier that x plus 1 is t,"},{"Start":"01:05.060 ","End":"01:06.815","Text":"so that\u0027s t squared plus 4."},{"Start":"01:06.815 ","End":"01:09.665","Text":"The minus 7 is minus 7 and x,"},{"Start":"01:09.665 ","End":"01:11.720","Text":"if we do the reverse substitution,"},{"Start":"01:11.720 ","End":"01:14.960","Text":"t is x plus 1, then x is t minus 1."},{"Start":"01:14.960 ","End":"01:16.645","Text":"Just bring this 1 over here."},{"Start":"01:16.645 ","End":"01:19.550","Text":"Instead of x, I put t minus 1 and this is what we get."},{"Start":"01:19.550 ","End":"01:23.975","Text":"Simplifying this, we get 3t minus 3 minus 7."},{"Start":"01:23.975 ","End":"01:27.545","Text":"Now we have a denominator which is free from the middle term."},{"Start":"01:27.545 ","End":"01:29.735","Text":"Now the numerator has got 2 separate bits,"},{"Start":"01:29.735 ","End":"01:31.490","Text":"and I want to split it up into"},{"Start":"01:31.490 ","End":"01:34.580","Text":"2 separate integrals because each one works a different way."},{"Start":"01:34.580 ","End":"01:38.780","Text":"What I mean is I take t over t squared plus 4 and I got 3 of those,"},{"Start":"01:38.780 ","End":"01:42.470","Text":"and I\u0027ve got minus 10 of 1 over t squared plus 4."},{"Start":"01:42.470 ","End":"01:44.660","Text":"This has a formula which involves the natural log,"},{"Start":"01:44.660 ","End":"01:47.690","Text":"and this one has a formula involving the arctangent."},{"Start":"01:47.690 ","End":"01:50.330","Text":"But in order for the logarithmic formula to work,"},{"Start":"01:50.330 ","End":"01:52.970","Text":"I need the derivative of the denominator on"},{"Start":"01:52.970 ","End":"01:56.570","Text":"the numerator and I have it not quite because I want 2t,"},{"Start":"01:56.570 ","End":"02:01.100","Text":"but I only have t. We fix that with the usual trick of writing the 2 here,"},{"Start":"02:01.100 ","End":"02:03.200","Text":"but then compensating by dividing by it."},{"Start":"02:03.200 ","End":"02:04.790","Text":"We really haven\u0027t changed anything."},{"Start":"02:04.790 ","End":"02:06.800","Text":"The formulas that I mentioned are,"},{"Start":"02:06.800 ","End":"02:10.190","Text":"this one where we have the derivative over the function"},{"Start":"02:10.190 ","End":"02:14.075","Text":"giving the natural logarithm and 1 over x squared plus a constant,"},{"Start":"02:14.075 ","End":"02:15.500","Text":"it has to be a positive constant,"},{"Start":"02:15.500 ","End":"02:17.935","Text":"by the way, is 1 over and so on."},{"Start":"02:17.935 ","End":"02:19.625","Text":"What do we get in our case?"},{"Start":"02:19.625 ","End":"02:22.400","Text":"In our case, f is t squared plus 4,"},{"Start":"02:22.400 ","End":"02:24.350","Text":"that gives us the 3 over 2 that was here."},{"Start":"02:24.350 ","End":"02:27.020","Text":"The natural logarithm of t squared plus 4,"},{"Start":"02:27.020 ","End":"02:29.824","Text":"I don\u0027t need the absolute value because this thing\u0027s positive,"},{"Start":"02:29.824 ","End":"02:32.000","Text":"minus 10 was always here."},{"Start":"02:32.000 ","End":"02:33.440","Text":"Now C is 4,"},{"Start":"02:33.440 ","End":"02:35.795","Text":"so this is square root of 4 first of all,"},{"Start":"02:35.795 ","End":"02:39.265","Text":"and then we can change the square root of 4 to 2."},{"Start":"02:39.265 ","End":"02:40.670","Text":"What we get is this."},{"Start":"02:40.670 ","End":"02:44.150","Text":"But I also pushed in another step where instead of dt,"},{"Start":"02:44.150 ","End":"02:46.775","Text":"I substituted back the x plus 1."},{"Start":"02:46.775 ","End":"02:50.030","Text":"T was x plus 1 and I substitute it back."},{"Start":"02:50.030 ","End":"02:52.925","Text":"That\u0027s this from t x plus 1,"},{"Start":"02:52.925 ","End":"02:55.580","Text":"and then and also here t is x plus 1."},{"Start":"02:55.580 ","End":"02:57.170","Text":"The square root of 4 was 2."},{"Start":"02:57.170 ","End":"03:02.820","Text":"I also did simplification that 10 over 2 is 5. This is the answer."}],"ID":4464},{"Watched":false,"Name":"Exercise 17","Duration":"2m 53s","ChapterTopicVideoID":4456,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.610","Text":"We have to compute the integral of 5x plus 14 over x^2 plus 4x plus 20."},{"Start":"00:05.610 ","End":"00:07.755","Text":"The denominator is what\u0027s important."},{"Start":"00:07.755 ","End":"00:09.660","Text":"Normally, we try to factorize it,"},{"Start":"00:09.660 ","End":"00:12.072","Text":"but if you try, you\u0027ll find that this is irreducible,"},{"Start":"00:12.072 ","End":"00:13.095","Text":"has no roots,"},{"Start":"00:13.095 ","End":"00:18.705","Text":"and so we try to get rid of the middle term by using the complete the square method."},{"Start":"00:18.705 ","End":"00:22.860","Text":"What I mean is we use the formula that\u0027s too familiar by now"},{"Start":"00:22.860 ","End":"00:27.108","Text":"and we apply it to x^2 plus 4x, where b is 4,"},{"Start":"00:27.108 ","End":"00:31.125","Text":"and we get that x^2 plus 4x is x plus 2 squared minus 4."},{"Start":"00:31.125 ","End":"00:32.340","Text":"If you have any doubt,"},{"Start":"00:32.340 ","End":"00:34.455","Text":"just expand this and you\u0027ll see you get this."},{"Start":"00:34.455 ","End":"00:36.600","Text":"Of course, we have a plus 20 here also,"},{"Start":"00:36.600 ","End":"00:40.590","Text":"and so we\u0027re going to end up with this minus 4 plus 20 is 16."},{"Start":"00:40.590 ","End":"00:42.645","Text":"We can rewrite this as this,"},{"Start":"00:42.645 ","End":"00:45.270","Text":"x plus 2 squared plus 16 on the bottom."},{"Start":"00:45.270 ","End":"00:47.895","Text":"So far, we\u0027ve just done algebra, no integration."},{"Start":"00:47.895 ","End":"00:50.205","Text":"Now, the next step is the substitution."},{"Start":"00:50.205 ","End":"00:53.190","Text":"x plus 2, we\u0027ll put as some variable."},{"Start":"00:53.190 ","End":"00:55.230","Text":"Usually, t is what we like,"},{"Start":"00:55.230 ","End":"00:56.850","Text":"and dt is dx."},{"Start":"00:56.850 ","End":"01:00.480","Text":"After the substitution, we get this. I need to explain."},{"Start":"01:00.480 ","End":"01:01.560","Text":"The denominator is clear,"},{"Start":"01:01.560 ","End":"01:04.350","Text":"x plus 2 is t, so t squared plus 16."},{"Start":"01:04.350 ","End":"01:05.685","Text":"The 14 is clear."},{"Start":"01:05.685 ","End":"01:08.940","Text":"Now the x, we need the reverse substitution."},{"Start":"01:08.940 ","End":"01:10.988","Text":"In other words, we need x in terms of t."},{"Start":"01:10.988 ","End":"01:12.180","Text":"That\u0027s easy enough."},{"Start":"01:12.180 ","End":"01:14.130","Text":"If you take this 2 over to the other side,"},{"Start":"01:14.130 ","End":"01:16.140","Text":"we get that x is t minus 2."},{"Start":"01:16.140 ","End":"01:18.735","Text":"That\u0027s the t minus 2 here. That\u0027s what x is."},{"Start":"01:18.735 ","End":"01:20.400","Text":"Now a little bit of algebra."},{"Start":"01:20.400 ","End":"01:25.245","Text":"5t minus 10 plus 14 is 5t plus 4,"},{"Start":"01:25.245 ","End":"01:27.375","Text":"and we get this expression."},{"Start":"01:27.375 ","End":"01:31.290","Text":"Now, here, the good thing to do is to split it up into 2 bits."},{"Start":"01:31.290 ","End":"01:33.825","Text":"A bit with t over this denominator"},{"Start":"01:33.825 ","End":"01:35.280","Text":"and the bit with 1 over."},{"Start":"01:35.280 ","End":"01:37.110","Text":"We get 5 of these and 4 of those."},{"Start":"01:37.110 ","End":"01:40.500","Text":"In other words, we get 5t over the denominator"},{"Start":"01:40.500 ","End":"01:41.778","Text":"and 4 over the denominator,"},{"Start":"01:41.778 ","End":"01:44.145","Text":"and then take the constants outside."},{"Start":"01:44.145 ","End":"01:45.495","Text":"This is what we reach."},{"Start":"01:45.495 ","End":"01:48.313","Text":"This one will work using the formula"},{"Start":"01:48.313 ","End":"01:51.195","Text":"where we have the derivative of the denominator on the top,"},{"Start":"01:51.195 ","End":"01:52.980","Text":"but we have to slightly adjust it."},{"Start":"01:52.980 ","End":"01:55.350","Text":"If I want the derivative of this to be here,"},{"Start":"01:55.350 ","End":"01:56.970","Text":"I need that to be a 2t,"},{"Start":"01:56.970 ","End":"01:59.640","Text":"so we use our usual trick where we put the 2 here"},{"Start":"01:59.640 ","End":"02:01.905","Text":"but then we compensate by writing the 2 here."},{"Start":"02:01.905 ","End":"02:04.680","Text":"Now we use 2 separate formulas for each bit."},{"Start":"02:04.680 ","End":"02:08.190","Text":"We use the bit about the natural logarithm for this one"},{"Start":"02:08.190 ","End":"02:10.752","Text":"so we get the natural logarithm of the denominator,"},{"Start":"02:10.752 ","End":"02:13.123","Text":"and for this, we use the arctangent formula"},{"Start":"02:13.123 ","End":"02:15.390","Text":"where it was expressed in terms of c,"},{"Start":"02:15.390 ","End":"02:17.130","Text":"1 over the square root of c"},{"Start":"02:17.130 ","End":"02:18.510","Text":"and t over the square root of c,"},{"Start":"02:18.510 ","End":"02:20.145","Text":"where c is this term here."},{"Start":"02:20.145 ","End":"02:21.630","Text":"This is what we end up with."},{"Start":"02:21.630 ","End":"02:24.780","Text":"Of course, 16 has a square root which is 4"},{"Start":"02:24.780 ","End":"02:27.420","Text":"so I\u0027m going to replace square root of 16 by 4,"},{"Start":"02:27.420 ","End":"02:29.640","Text":"and I\u0027ll do one other step at the same time."},{"Start":"02:29.640 ","End":"02:34.560","Text":"I\u0027ll substitute back instead of t, I\u0027ll write x plus 2 the way it was."},{"Start":"02:34.560 ","End":"02:40.260","Text":"We end up by getting this 5 over 2 just as a decimal 2.5,"},{"Start":"02:40.260 ","End":"02:43.365","Text":"and t is x plus 2, and so on."},{"Start":"02:43.365 ","End":"02:45.540","Text":"The square root of 16 is 4."},{"Start":"02:45.540 ","End":"02:47.820","Text":"Here, the 4 cancels with this 4,"},{"Start":"02:47.820 ","End":"02:49.950","Text":"which is why there\u0027s no constant here."},{"Start":"02:49.950 ","End":"02:54.760","Text":"Of course, the c for the integration constant, and we\u0027re done."}],"ID":4465},{"Watched":false,"Name":"Exercise 18","Duration":"2m 49s","ChapterTopicVideoID":4457,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"Here we want to compute the following integral,"},{"Start":"00:02.340 ","End":"00:05.520","Text":"x squared plus x minus 1/x cubed minus x."},{"Start":"00:05.520 ","End":"00:07.575","Text":"Always look at the denominator first,"},{"Start":"00:07.575 ","End":"00:08.880","Text":"try to factorize it."},{"Start":"00:08.880 ","End":"00:11.010","Text":"Well, clearly I can take x out"},{"Start":"00:11.010 ","End":"00:13.710","Text":"of the brackets and I\u0027ll end up"},{"Start":"00:13.710 ","End":"00:17.115","Text":"with x times x squared minus 1."},{"Start":"00:17.115 ","End":"00:19.140","Text":"But that\u0027s not all, x squared minus 1"},{"Start":"00:19.140 ","End":"00:20.910","Text":"is 1 of those difference of squares,"},{"Start":"00:20.910 ","End":"00:23.160","Text":"so we can factorize that also"},{"Start":"00:23.160 ","End":"00:25.305","Text":"into x minus 1 x plus 1."},{"Start":"00:25.305 ","End":"00:27.800","Text":"Already, we have 3 linear factors,"},{"Start":"00:27.800 ","End":"00:30.860","Text":"x minus 0, x minus 1, x plus 1."},{"Start":"00:30.860 ","End":"00:33.290","Text":"The word partial fractions should come"},{"Start":"00:33.290 ","End":"00:35.465","Text":"to mind so we can decompose this."},{"Start":"00:35.465 ","End":"00:37.670","Text":"Presumably, this rational expression"},{"Start":"00:37.670 ","End":"00:39.380","Text":"was reached by combining"},{"Start":"00:39.380 ","End":"00:40.580","Text":"something over x,"},{"Start":"00:40.580 ","End":"00:41.900","Text":"something over x minus 1,"},{"Start":"00:41.900 ","End":"00:43.140","Text":"something over x plus 1."},{"Start":"00:43.140 ","End":"00:45.050","Text":"We have the reverse task of finding out"},{"Start":"00:45.050 ","End":"00:46.490","Text":"what A, B, and C are."},{"Start":"00:46.490 ","End":"00:48.050","Text":"The usual method,"},{"Start":"00:48.050 ","End":"00:49.250","Text":"we put everything over a"},{"Start":"00:49.250 ","End":"00:50.890","Text":"common denominator which is this."},{"Start":"00:50.890 ","End":"00:52.220","Text":"But then we throw out the common"},{"Start":"00:52.220 ","End":"00:53.800","Text":"denominator from both sides,"},{"Start":"00:53.800 ","End":"00:55.850","Text":"and what we\u0027re left with is this thing"},{"Start":"00:55.850 ","End":"00:59.660","Text":"as is and A times the 2 missing terms,"},{"Start":"00:59.660 ","End":"01:01.175","Text":"which is this and this."},{"Start":"01:01.175 ","End":"01:04.340","Text":"B times this times this here"},{"Start":"01:04.340 ","End":"01:06.680","Text":"and C times this times this."},{"Start":"01:06.680 ","End":"01:09.420","Text":"This is not really just an inequality,"},{"Start":"01:09.420 ","End":"01:10.669","Text":"I mean this is an identity,"},{"Start":"01:10.669 ","End":"01:12.530","Text":"meaning it\u0027s true for all x."},{"Start":"01:12.530 ","End":"01:14.060","Text":"We can substitute whatever"},{"Start":"01:14.060 ","End":"01:15.715","Text":"values of x we like."},{"Start":"01:15.715 ","End":"01:19.130","Text":"Let\u0027s try x equals 0 because that will make"},{"Start":"01:19.130 ","End":"01:22.610","Text":"this term and this term 0, 0 and the 0."},{"Start":"01:22.610 ","End":"01:25.280","Text":"All we\u0027re left with is minus 1 equals A"},{"Start":"01:25.280 ","End":"01:28.855","Text":"times minus 1 and that gives us that A is 1."},{"Start":"01:28.855 ","End":"01:30.965","Text":"Let\u0027s try to make something else 0."},{"Start":"01:30.965 ","End":"01:33.740","Text":"Let\u0027s try to set x equals 1"},{"Start":"01:33.740 ","End":"01:35.780","Text":"to get this 0 and this 0."},{"Start":"01:35.780 ","End":"01:37.160","Text":"If I let x equals 1,"},{"Start":"01:37.160 ","End":"01:39.030","Text":"I get the following expression,"},{"Start":"01:39.030 ","End":"01:41.340","Text":"where only this 1 is non-zero."},{"Start":"01:41.340 ","End":"01:44.580","Text":"We have B times 1 times 2 which is 2 B,"},{"Start":"01:44.580 ","End":"01:46.410","Text":"and here we\u0027re left with"},{"Start":"01:46.410 ","End":"01:48.150","Text":"1 plus 1 minus 1 is 1,"},{"Start":"01:48.150 ","End":"01:49.725","Text":"so 2B is 1,"},{"Start":"01:49.725 ","End":"01:51.555","Text":"which means that B is a half."},{"Start":"01:51.555 ","End":"01:53.630","Text":"Finally, I let x equals minus 1"},{"Start":"01:53.630 ","End":"01:55.820","Text":"to let the x plus 1 things be 0."},{"Start":"01:55.820 ","End":"01:57.590","Text":"We get a 0 here and here,"},{"Start":"01:57.590 ","End":"01:59.480","Text":"and on this side, we get minus 1"},{"Start":"01:59.480 ","End":"02:01.370","Text":"and this side we get 2C,"},{"Start":"02:01.370 ","End":"02:04.100","Text":"which gives us that C is minus 1/2."},{"Start":"02:04.100 ","End":"02:05.120","Text":"Which means that I can"},{"Start":"02:05.120 ","End":"02:08.060","Text":"rewrite this with these constants,"},{"Start":"02:08.060 ","End":"02:10.100","Text":"and put them under the integral,"},{"Start":"02:10.100 ","End":"02:13.040","Text":"we get 1, which is A/x."},{"Start":"02:13.040 ","End":"02:15.080","Text":"This 1/2 over x minus 1,"},{"Start":"02:15.080 ","End":"02:17.045","Text":"this minus 1/2 over x plus 1."},{"Start":"02:17.045 ","End":"02:20.255","Text":"We have to separate the constants out."},{"Start":"02:20.255 ","End":"02:22.190","Text":"This is 1/2, this is minus 1/2,"},{"Start":"02:22.190 ","End":"02:23.240","Text":"this is just as is,"},{"Start":"02:23.240 ","End":"02:24.870","Text":"then apply the rule for that"},{"Start":"02:24.870 ","End":"02:25.910","Text":"if we have the derivative"},{"Start":"02:25.910 ","End":"02:26.870","Text":"of the bottom on the top,"},{"Start":"02:26.870 ","End":"02:28.745","Text":"it\u0027s natural logarithm of the bottom."},{"Start":"02:28.745 ","End":"02:31.565","Text":"Here it is the formula that I mentioned"},{"Start":"02:31.565 ","End":"02:32.815","Text":"is what we\u0027re going to use,"},{"Start":"02:32.815 ","End":"02:35.450","Text":"because 1 is the derivative of x plus A,"},{"Start":"02:35.450 ","End":"02:37.430","Text":"so we get natural logarithm of x plus A"},{"Start":"02:37.430 ","End":"02:41.285","Text":"and in 3 times and we get natural log first of all of x,"},{"Start":"02:41.285 ","End":"02:42.820","Text":"then of x minus 1,"},{"Start":"02:42.820 ","End":"02:44.990","Text":"then of x plus 1 and not forget"},{"Start":"02:44.990 ","End":"02:47.030","Text":"the constants that were the 1/2 here,"},{"Start":"02:47.030 ","End":"02:48.080","Text":"and the minus 1/2 here,"},{"Start":"02:48.080 ","End":"02:50.250","Text":"and this is the answer."}],"ID":4466},{"Watched":false,"Name":"Exercise 19","Duration":"3m 39s","ChapterTopicVideoID":4458,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.890","Text":"In this exercise, we have to compute"},{"Start":"00:01.890 ","End":"00:03.195","Text":"the following integral,"},{"Start":"00:03.195 ","End":"00:05.100","Text":"6x squared plus 4x minus 6"},{"Start":"00:05.100 ","End":"00:08.535","Text":"over x cubed minus 7x minus 6."},{"Start":"00:08.535 ","End":"00:09.870","Text":"I like to sometimes check"},{"Start":"00:09.870 ","End":"00:11.700","Text":"that the degrees are okay."},{"Start":"00:11.700 ","End":"00:15.060","Text":"The degree in the numerator is equal to 2,"},{"Start":"00:15.060 ","End":"00:17.759","Text":"and the degree in the denominator is 3,"},{"Start":"00:17.759 ","End":"00:20.370","Text":"and as long as 2 is less than 3,"},{"Start":"00:20.370 ","End":"00:21.495","Text":"then we\u0027re okay."},{"Start":"00:21.495 ","End":"00:22.530","Text":"The next step is to try"},{"Start":"00:22.530 ","End":"00:25.200","Text":"and factorize the denominator."},{"Start":"00:25.200 ","End":"00:27.809","Text":"There\u0027s no easy way in general"},{"Start":"00:27.809 ","End":"00:30.390","Text":"of solving a cubic equation."},{"Start":"00:30.390 ","End":"00:32.700","Text":"But there are some techniques"},{"Start":"00:32.700 ","End":"00:34.350","Text":"involving a bit of guesswork."},{"Start":"00:34.350 ","End":"00:35.960","Text":"In this case, let\u0027s assume"},{"Start":"00:35.960 ","End":"00:37.440","Text":"that it has 3 factors,"},{"Start":"00:37.440 ","End":"00:38.960","Text":"that it completely factorizes,"},{"Start":"00:38.960 ","End":"00:40.550","Text":"which will likely be the case"},{"Start":"00:40.550 ","End":"00:42.215","Text":"on an exam question."},{"Start":"00:42.215 ","End":"00:44.360","Text":"If it has 3 roots,"},{"Start":"00:44.360 ","End":"00:45.905","Text":"x^1, x^2, and x^3,"},{"Start":"00:45.905 ","End":"00:48.290","Text":"I want to write this as x minus x^1,"},{"Start":"00:48.290 ","End":"00:50.011","Text":"x minus x^2 x minus x^3."},{"Start":"00:50.011 ","End":"00:51.440","Text":"But I don\u0027t know what these"},{"Start":"00:51.440 ","End":"00:53.120","Text":"x^1, x^2, x^3 are."},{"Start":"00:53.120 ","End":"00:55.030","Text":"However, there are certain rules of thumb."},{"Start":"00:55.030 ","End":"00:57.050","Text":"For example, if the coefficient here is 1"},{"Start":"00:57.050 ","End":"00:58.490","Text":"and it\u0027s all whole numbers,"},{"Start":"00:58.490 ","End":"01:02.825","Text":"then each of these will be a factor of minus 6."},{"Start":"01:02.825 ","End":"01:05.450","Text":"What we can say is that all the factors"},{"Start":"01:05.450 ","End":"01:09.335","Text":"of minus 6 plus or minus 1,"},{"Start":"01:09.335 ","End":"01:11.864","Text":"plus or minus 2, plus or minus 3,"},{"Start":"01:11.864 ","End":"01:13.325","Text":"and plus or minus 6."},{"Start":"01:13.325 ","End":"01:15.650","Text":"I have really 8 possibilities"},{"Start":"01:15.650 ","End":"01:17.180","Text":"for each of the roots."},{"Start":"01:17.180 ","End":"01:19.190","Text":"Now, just by trial and error,"},{"Start":"01:19.190 ","End":"01:21.290","Text":"by substituting each of these 8 things"},{"Start":"01:21.290 ","End":"01:24.140","Text":"into the equation and looking for a 0,"},{"Start":"01:24.140 ","End":"01:26.225","Text":"what we get is that 3 of them work,"},{"Start":"01:26.225 ","End":"01:28.235","Text":"and the 3 that work are minus 1,"},{"Start":"01:28.235 ","End":"01:29.910","Text":"minus 2 and 3."},{"Start":"01:29.910 ","End":"01:33.275","Text":"If I put x^1 is minus 1 here and so on,"},{"Start":"01:33.275 ","End":"01:35.030","Text":"what I\u0027m left with is that"},{"Start":"01:35.030 ","End":"01:37.730","Text":"this factorizes into x plus 1,"},{"Start":"01:37.730 ","End":"01:41.135","Text":"x plus 2, and x minus 3."},{"Start":"01:41.135 ","End":"01:43.580","Text":"That factorizes the denominator,"},{"Start":"01:43.580 ","End":"01:45.890","Text":"so I rewrite the integral like this,"},{"Start":"01:45.890 ","End":"01:47.600","Text":"and now the term partial fractions"},{"Start":"01:47.600 ","End":"01:48.825","Text":"comes to mind."},{"Start":"01:48.825 ","End":"01:50.240","Text":"We assume that someone"},{"Start":"01:50.240 ","End":"01:52.790","Text":"got to this expression by putting"},{"Start":"01:52.790 ","End":"01:53.930","Text":"a common denominator"},{"Start":"01:53.930 ","End":"01:55.355","Text":"for an expression like this,"},{"Start":"01:55.355 ","End":"01:57.320","Text":"A over x plus 1, B over x plus 2,"},{"Start":"01:57.320 ","End":"01:58.730","Text":"C over x minus 3."},{"Start":"01:58.730 ","End":"02:01.130","Text":"The way to reach these A, B, and C"},{"Start":"02:01.130 ","End":"02:02.990","Text":"is to again, put a common"},{"Start":"02:02.990 ","End":"02:04.655","Text":"denominator of this,"},{"Start":"02:04.655 ","End":"02:05.540","Text":"but then we can throw out"},{"Start":"02:05.540 ","End":"02:06.770","Text":"the common denominator."},{"Start":"02:06.770 ","End":"02:08.090","Text":"What we\u0027re left with is that,"},{"Start":"02:08.090 ","End":"02:10.370","Text":"this thing as A is equal to A"},{"Start":"02:10.370 ","End":"02:12.589","Text":"times the 2 missing factors,"},{"Start":"02:12.589 ","End":"02:13.640","Text":"this and this, and so on"},{"Start":"02:13.640 ","End":"02:15.380","Text":"with B times this and this,"},{"Start":"02:15.380 ","End":"02:17.570","Text":"and C times this and this."},{"Start":"02:17.570 ","End":"02:19.700","Text":"Next thing to do is to substitute values."},{"Start":"02:19.700 ","End":"02:21.320","Text":"You can substitute any values."},{"Start":"02:21.320 ","End":"02:23.300","Text":"But it\u0027s better to choose something that"},{"Start":"02:23.300 ","End":"02:25.295","Text":"will make some of the terms disappear."},{"Start":"02:25.295 ","End":"02:27.080","Text":"Like if x is minus 1,"},{"Start":"02:27.080 ","End":"02:29.440","Text":"then both of these x plus 1\u0027s disappear,"},{"Start":"02:29.440 ","End":"02:30.865","Text":"and give a 0 here."},{"Start":"02:30.865 ","End":"02:32.300","Text":"All we\u0027re left with is,"},{"Start":"02:32.300 ","End":"02:34.160","Text":"here we get minus 4,"},{"Start":"02:34.160 ","End":"02:36.379","Text":"and here we get minus 4,"},{"Start":"02:36.379 ","End":"02:38.060","Text":"and so we get that minus 4A"},{"Start":"02:38.060 ","End":"02:40.070","Text":"is minus 4 and A is 1."},{"Start":"02:40.070 ","End":"02:43.130","Text":"Then let\u0027s try substituting x is minus 2,"},{"Start":"02:43.130 ","End":"02:45.215","Text":"which will make this 0 and this 0."},{"Start":"02:45.215 ","End":"02:46.370","Text":"That\u0027s this and this."},{"Start":"02:46.370 ","End":"02:49.325","Text":"All we\u0027re left with is 5B equals 10,"},{"Start":"02:49.325 ","End":"02:51.245","Text":"and then B equals 2,"},{"Start":"02:51.245 ","End":"02:52.820","Text":"and finally, x equals 3"},{"Start":"02:52.820 ","End":"02:54.740","Text":"to get rid of this and this."},{"Start":"02:54.740 ","End":"02:56.120","Text":"These 2 are 0 \u0027s,"},{"Start":"02:56.120 ","End":"02:57.530","Text":"and if you plug in,"},{"Start":"02:57.530 ","End":"02:59.900","Text":"we get 60C times 20,"},{"Start":"02:59.900 ","End":"03:01.895","Text":"which makes C equals 3."},{"Start":"03:01.895 ","End":"03:03.920","Text":"Now, we can substitute"},{"Start":"03:03.920 ","End":"03:05.660","Text":"instead of this, this,"},{"Start":"03:05.660 ","End":"03:07.970","Text":"but with A, B, and C as found,"},{"Start":"03:07.970 ","End":"03:09.845","Text":"and so we end up with this."},{"Start":"03:09.845 ","End":"03:11.480","Text":"Our original integral is equal"},{"Start":"03:11.480 ","End":"03:13.519","Text":"to the sum of 3 separate integrals."},{"Start":"03:13.519 ","End":"03:16.129","Text":"Let\u0027s split this up into 3 bits."},{"Start":"03:16.129 ","End":"03:18.620","Text":"But we\u0027ll also take the constants out."},{"Start":"03:18.620 ","End":"03:21.330","Text":"We get this plus twice,"},{"Start":"03:21.330 ","End":"03:22.745","Text":"1 over x plus 2,"},{"Start":"03:22.745 ","End":"03:24.785","Text":"3 times 1 over x minus 3."},{"Start":"03:24.785 ","End":"03:26.570","Text":"Then we just use the formula"},{"Start":"03:26.570 ","End":"03:28.580","Text":"with the logarithm of the denominator"},{"Start":"03:28.580 ","End":"03:30.050","Text":"as the numerator in each case"},{"Start":"03:30.050 ","End":"03:31.760","Text":"is the derivative of the denominator."},{"Start":"03:31.760 ","End":"03:34.010","Text":"We get natural log of this and this,"},{"Start":"03:34.010 ","End":"03:37.280","Text":"and this added together with the constants"},{"Start":"03:37.280 ","End":"03:38.430","Text":"and absolute value."},{"Start":"03:38.430 ","End":"03:40.600","Text":"The answer."}],"ID":4467},{"Watched":false,"Name":"Exercise 20","Duration":"4m 23s","ChapterTopicVideoID":4459,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.940","Text":"Here we have to compute the integral of 10x over x^4 minus 13x^2 plus 36,"},{"Start":"00:05.940 ","End":"00:09.060","Text":"degree 1 over degree 4, the denominator."},{"Start":"00:09.060 ","End":"00:11.205","Text":"Let\u0027s see if we can factorize it."},{"Start":"00:11.205 ","End":"00:15.480","Text":"Ideally, what I\u0027d like to get is complete factorization"},{"Start":"00:15.480 ","End":"00:19.980","Text":"would be to have it as x minus x_1 up to x_4,"},{"Start":"00:19.980 ","End":"00:23.190","Text":"where these x_1 to x_4 are the roots."},{"Start":"00:23.190 ","End":"00:24.990","Text":"Now, fortunately, this one,"},{"Start":"00:24.990 ","End":"00:27.740","Text":"if you look at it, it doesn\u0027t have any even powers."},{"Start":"00:27.740 ","End":"00:29.855","Text":"There\u0027s no x^3 and there\u0027s no x."},{"Start":"00:29.855 ","End":"00:33.200","Text":"When this happens, it\u0027s very useful to substitute"},{"Start":"00:33.200 ","End":"00:36.920","Text":"x^2 is equal to t because then I get something much simpler."},{"Start":"00:36.920 ","End":"00:39.290","Text":"This x^4 becomes t^2,"},{"Start":"00:39.290 ","End":"00:42.850","Text":"and then this becomes 13t, instead of 13x^2,"},{"Start":"00:42.850 ","End":"00:44.720","Text":"and 36 is 0."},{"Start":"00:44.720 ","End":"00:46.460","Text":"Now this is a quadratic."},{"Start":"00:46.460 ","End":"00:50.335","Text":"If it\u0027s a quadratic, then we can find its solutions, and"},{"Start":"00:50.335 ","End":"00:54.875","Text":"it\u0027s easy enough to get that it has 2 solutions, 9 and 4."},{"Start":"00:54.875 ","End":"00:57.620","Text":"But the 9 and 4 are values of t."},{"Start":"00:57.620 ","End":"01:00.728","Text":"So if t is 9, then x^2 is 9,"},{"Start":"01:00.728 ","End":"01:03.665","Text":"and if t is 4, then x^2 is 4,"},{"Start":"01:03.665 ","End":"01:09.114","Text":"and that gives us that x_1 and x_2 are plus and minus 3,"},{"Start":"01:09.114 ","End":"01:15.175","Text":"and the other 2 x\u0027s, x_3 and x_4, column that\u0027ll be plus and minus 2 square root of 4."},{"Start":"01:15.175 ","End":"01:20.360","Text":"So now we have 4 roots and then we can write this quartic equation,"},{"Start":"01:20.360 ","End":"01:24.050","Text":"power 4 equation, as completely factorized like this."},{"Start":"01:24.050 ","End":"01:29.097","Text":"Now we can write the original integral of this with the denominator replaced"},{"Start":"01:29.097 ","End":"01:33.695","Text":"by its factorization and partial fractions is next, of course."},{"Start":"01:33.695 ","End":"01:36.365","Text":"So we have these 4 factors, each linear."},{"Start":"01:36.365 ","End":"01:40.212","Text":"So we get a constant over each of the linear A, B, C, and D."},{"Start":"01:40.212 ","End":"01:43.625","Text":"Let\u0027s, first of all, put both sides over a common denominator,"},{"Start":"01:43.625 ","End":"01:45.630","Text":"which is the product of all these,"},{"Start":"01:45.630 ","End":"01:48.847","Text":"and what we get here, the 10x stays 10x,"},{"Start":"01:48.847 ","End":"01:54.710","Text":"and here for the common denominator, I multiply each constant by the 3 remaining factors."},{"Start":"01:54.710 ","End":"01:56.945","Text":"A is multiplied by this, this, and this."},{"Start":"01:56.945 ","End":"01:59.510","Text":"B is multiplied by this, this, and this,"},{"Start":"01:59.510 ","End":"02:03.155","Text":"and so on, and we get this ugly-looking expression."},{"Start":"02:03.155 ","End":"02:06.350","Text":"But notice that each of the factors of p is 3 times."},{"Start":"02:06.350 ","End":"02:10.640","Text":"For example, x minus 3 appears here and here and here."},{"Start":"02:10.640 ","End":"02:12.440","Text":"So if I let x equals 3,"},{"Start":"02:12.440 ","End":"02:13.955","Text":"then it would get rid of those."},{"Start":"02:13.955 ","End":"02:18.590","Text":"Anyway, let\u0027s start with x equals minus 3,"},{"Start":"02:18.590 ","End":"02:21.635","Text":"and that will get rid of this one, this one, and this one."},{"Start":"02:21.635 ","End":"02:23.915","Text":"See the 0 here, the 0 here, the 0 here."},{"Start":"02:23.915 ","End":"02:26.890","Text":"All we have is on the left, we get minus 30,"},{"Start":"02:26.890 ","End":"02:28.980","Text":"and if you compute this,"},{"Start":"02:28.980 ","End":"02:32.724","Text":"we also get minus 30 so ultimately A is equal to 1"},{"Start":"02:32.724 ","End":"02:35.540","Text":"because minus 30 is minus 30A."},{"Start":"02:35.540 ","End":"02:37.430","Text":"Then we\u0027ll try x equals 3."},{"Start":"02:37.430 ","End":"02:39.805","Text":"Like I said, we\u0027ll get rid of 3 of these."},{"Start":"02:39.805 ","End":"02:43.410","Text":"So they have a 0, a 0, and a 0, and this thing,"},{"Start":"02:43.410 ","End":"02:46.010","Text":"if you let x equals 3, but I\u0027ll tell you,"},{"Start":"02:46.010 ","End":"02:47.960","Text":"for example, 3 plus 3 is 6,"},{"Start":"02:47.960 ","End":"02:50.270","Text":"3 plus 2 is 5, 3 minus 2 is 1,"},{"Start":"02:50.270 ","End":"02:52.777","Text":"6 times 5 times 1 is 30, 30B,"},{"Start":"02:52.777 ","End":"02:53.945","Text":"and on the left,"},{"Start":"02:53.945 ","End":"02:58.160","Text":"3 times 10 is 30 so 30 equals 30B, and B equals 1."},{"Start":"02:58.160 ","End":"03:00.170","Text":"Similarly, using the same trick,"},{"Start":"03:00.170 ","End":"03:03.740","Text":"we let minus 2, this, this, and this disappear."},{"Start":"03:03.740 ","End":"03:07.255","Text":"That\u0027s the three 0s here and we get that C is 1."},{"Start":"03:07.255 ","End":"03:10.970","Text":"At last, you let x equals 2, we get minus 20."},{"Start":"03:10.970 ","End":"03:15.065","Text":"D is equal to 20, so D is minus 1."},{"Start":"03:15.065 ","End":"03:19.520","Text":"Finally, we find all 4 constants and we just plug them in,"},{"Start":"03:19.520 ","End":"03:21.710","Text":"and we plug them in to this line here,"},{"Start":"03:21.710 ","End":"03:23.270","Text":"we put in this expression"},{"Start":"03:23.270 ","End":"03:25.370","Text":"but with the A, B, and C, and D replaced."},{"Start":"03:25.370 ","End":"03:29.840","Text":"So it will look like this is our integral and get to this stage,"},{"Start":"03:29.840 ","End":"03:31.970","Text":"just separate them into 4 bits,"},{"Start":"03:31.970 ","End":"03:36.170","Text":"but also take the minus 1 outside the integral sign to have the integral of this,"},{"Start":"03:36.170 ","End":"03:38.090","Text":"integral of this, minus the integral of this,"},{"Start":"03:38.090 ","End":"03:39.455","Text":"minus the integral of this."},{"Start":"03:39.455 ","End":"03:42.830","Text":"This is a standard case of natural log of the denominator."},{"Start":"03:42.830 ","End":"03:46.280","Text":"Remember that the integral of 1 over x plus a number is"},{"Start":"03:46.280 ","End":"03:50.300","Text":"a natural log of the denominator in absolute value, strictly speaking."},{"Start":"03:50.300 ","End":"03:55.325","Text":"So we apply it 4 times to each of these and we get the following expression."},{"Start":"03:55.325 ","End":"03:59.015","Text":"Now what we can do is because of the laws of logarithms,"},{"Start":"03:59.015 ","End":"04:02.215","Text":"the sum of the logarithms is the logarithm of the product."},{"Start":"04:02.215 ","End":"04:06.425","Text":"So these two give us logarithm of this times this,"},{"Start":"04:06.425 ","End":"04:09.710","Text":"and we have a minus, it\u0027s a division, so it goes into the denominator."},{"Start":"04:09.710 ","End":"04:13.242","Text":"I can take all these four and put the pluses into the numerator"},{"Start":"04:13.242 ","End":"04:15.278","Text":"and the minuses into the denominator,"},{"Start":"04:15.278 ","End":"04:17.120","Text":"and if I multiply this out,"},{"Start":"04:17.120 ","End":"04:18.961","Text":"I have a fairly simple expression."},{"Start":"04:18.961 ","End":"04:24.720","Text":"It\u0027s a natural logarithm of x^2 minus 9 over x^2 minus 4 plus the constant."}],"ID":4468},{"Watched":false,"Name":"Exercise 21","Duration":"3m 19s","ChapterTopicVideoID":4460,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.860","Text":"Here, we have to solve this integral,"},{"Start":"00:01.860 ","End":"00:03.935","Text":"8x over something."},{"Start":"00:03.935 ","End":"00:05.310","Text":"This denominator is already"},{"Start":"00:05.310 ","End":"00:07.005","Text":"completely factorized for us,"},{"Start":"00:07.005 ","End":"00:09.330","Text":"x minus 2 squared times x plus 2."},{"Start":"00:09.330 ","End":"00:10.470","Text":"So we can jump straight"},{"Start":"00:10.470 ","End":"00:12.428","Text":"to the partial fractions part,"},{"Start":"00:12.428 ","End":"00:13.890","Text":"and write this as,"},{"Start":"00:13.890 ","End":"00:15.630","Text":"because of the x minus 2 squared,"},{"Start":"00:15.630 ","End":"00:17.100","Text":"we need both an x minus 2"},{"Start":"00:17.100 ","End":"00:18.795","Text":"and an x minus 2 squared."},{"Start":"00:18.795 ","End":"00:20.430","Text":"For the x plus 2, we only need"},{"Start":"00:20.430 ","End":"00:22.500","Text":"a representative of x plus 2."},{"Start":"00:22.500 ","End":"00:25.080","Text":"We need all the powers up to 2 here"},{"Start":"00:25.080 ","End":"00:26.370","Text":"and up to 1 here,"},{"Start":"00:26.370 ","End":"00:27.750","Text":"so that\u0027s these 3 terms"},{"Start":"00:27.750 ","End":"00:29.685","Text":"with a constant on top of each."},{"Start":"00:29.685 ","End":"00:31.020","Text":"We imagine that someone"},{"Start":"00:31.020 ","End":"00:32.141","Text":"started out with this"},{"Start":"00:32.141 ","End":"00:34.470","Text":"and put a common denominator to get this,"},{"Start":"00:34.470 ","End":"00:35.955","Text":"and we\u0027re doing the reverse."},{"Start":"00:35.955 ","End":"00:37.830","Text":"Again, we put a common denominator,"},{"Start":"00:37.830 ","End":"00:38.985","Text":"which is this thing,"},{"Start":"00:38.985 ","End":"00:41.450","Text":"and then we throw out the denominators"},{"Start":"00:41.450 ","End":"00:42.980","Text":"and we compare the numerators."},{"Start":"00:42.980 ","End":"00:43.940","Text":"Here we get 8x,"},{"Start":"00:43.940 ","End":"00:46.430","Text":"and here we have a times"},{"Start":"00:46.430 ","End":"00:47.390","Text":"whatever\u0027s missing,"},{"Start":"00:47.390 ","End":"00:49.025","Text":"we had an x minus 2 here,"},{"Start":"00:49.025 ","End":"00:50.450","Text":"so we need another x minus 2"},{"Start":"00:50.450 ","End":"00:51.575","Text":"and an x plus 2."},{"Start":"00:51.575 ","End":"00:53.480","Text":"For the B, we had x minus 2 squared,"},{"Start":"00:53.480 ","End":"00:55.400","Text":"so all we are missing is x plus 2."},{"Start":"00:55.400 ","End":"00:58.325","Text":"For C, we\u0027re missing the x minus 2 squared."},{"Start":"00:58.325 ","End":"01:00.920","Text":"Now, this thing is not really an equality."},{"Start":"01:00.920 ","End":"01:02.210","Text":"It\u0027s an identity which means"},{"Start":"01:02.210 ","End":"01:03.690","Text":"that for every x it will work."},{"Start":"01:03.690 ","End":"01:05.780","Text":"We plug in values of x"},{"Start":"01:05.780 ","End":"01:07.625","Text":"that are most useful to us,"},{"Start":"01:07.625 ","End":"01:09.440","Text":"so x equals 2,"},{"Start":"01:09.440 ","End":"01:11.090","Text":"we\u0027ll make this disappear,"},{"Start":"01:11.090 ","End":"01:13.129","Text":"and this disappear,"},{"Start":"01:13.129 ","End":"01:14.420","Text":"and all we\u0027ll be left with"},{"Start":"01:14.420 ","End":"01:17.780","Text":"is b times 4 equals 16."},{"Start":"01:17.780 ","End":"01:18.950","Text":"The other 2 are 0."},{"Start":"01:18.950 ","End":"01:21.080","Text":"This gives us that B is 4."},{"Start":"01:21.080 ","End":"01:22.850","Text":"The other useful thing to substitute"},{"Start":"01:22.850 ","End":"01:25.250","Text":"would be x equals minus 2,"},{"Start":"01:25.250 ","End":"01:28.340","Text":"because then we\u0027ll get a 0 here and here,"},{"Start":"01:28.340 ","End":"01:32.090","Text":"and we\u0027ll get 16 C equals minus 16,"},{"Start":"01:32.090 ","End":"01:32.870","Text":"you can check it,"},{"Start":"01:32.870 ","End":"01:35.470","Text":"and that gives us that C is minus 1."},{"Start":"01:35.470 ","End":"01:37.230","Text":"For the last value doesn\u0027t really matter,"},{"Start":"01:37.230 ","End":"01:39.380","Text":"I often choose 0 because it\u0027s easy to work with,"},{"Start":"01:39.380 ","End":"01:41.809","Text":"and so we get on this side 0,"},{"Start":"01:41.809 ","End":"01:43.625","Text":"and here we get a times,"},{"Start":"01:43.625 ","End":"01:45.140","Text":"this is minus 2 plus 2,"},{"Start":"01:45.140 ","End":"01:46.700","Text":"so it\u0027s minus 4."},{"Start":"01:46.700 ","End":"01:48.140","Text":"Here we get 2B,"},{"Start":"01:48.140 ","End":"01:51.590","Text":"and here we get minus 2 squared is 4 times C,"},{"Start":"01:51.590 ","End":"01:53.150","Text":"but we know what C and B"},{"Start":"01:53.150 ","End":"01:55.310","Text":"are already because we found them."},{"Start":"01:55.310 ","End":"01:57.665","Text":"C is minus 1 and B is 4."},{"Start":"01:57.665 ","End":"01:59.570","Text":"We get that A is 1."},{"Start":"01:59.570 ","End":"02:02.420","Text":"That means we can write the original integral,"},{"Start":"02:02.420 ","End":"02:04.470","Text":"split up into partial fractions,"},{"Start":"02:04.470 ","End":"02:05.730","Text":"we have A, B, and C,"},{"Start":"02:05.730 ","End":"02:07.425","Text":"and this is what we get."},{"Start":"02:07.425 ","End":"02:10.220","Text":"Break this up into 3 separate integrals"},{"Start":"02:10.220 ","End":"02:12.020","Text":"and also take the constants outside,"},{"Start":"02:12.020 ","End":"02:13.640","Text":"the minus 1 comes here as a minus,"},{"Start":"02:13.640 ","End":"02:15.455","Text":"the 4 is here, the 4."},{"Start":"02:15.455 ","End":"02:18.230","Text":"We need to use 2 separate formulas."},{"Start":"02:18.230 ","End":"02:21.320","Text":"In the case of just x minus 2 or x plus 2,"},{"Start":"02:21.320 ","End":"02:22.818","Text":"we use a natural logarithm,"},{"Start":"02:22.818 ","End":"02:24.425","Text":"and for all other exponents,"},{"Start":"02:24.425 ","End":"02:25.940","Text":"we need another formula."},{"Start":"02:25.940 ","End":"02:28.430","Text":"Basically, for x plus a in the denominator,"},{"Start":"02:28.430 ","End":"02:29.620","Text":"we get a natural logarithm,"},{"Start":"02:29.620 ","End":"02:30.740","Text":"and if it\u0027s any other power,"},{"Start":"02:30.740 ","End":"02:32.930","Text":"we use the formula that we raise"},{"Start":"02:32.930 ","End":"02:35.135","Text":"the power by 1 and divide by it."},{"Start":"02:35.135 ","End":"02:37.310","Text":"The x minus 2 and x plus 2"},{"Start":"02:37.310 ","End":"02:39.035","Text":"came out as logarithms."},{"Start":"02:39.035 ","End":"02:41.240","Text":"That\u0027s the log of x minus 2,"},{"Start":"02:41.240 ","End":"02:42.290","Text":"and there\u0027s a minus here,"},{"Start":"02:42.290 ","End":"02:44.090","Text":"and there\u0027s a log of the x plus 2."},{"Start":"02:44.090 ","End":"02:46.400","Text":"But for this term, it\u0027s to the power"},{"Start":"02:46.400 ","End":"02:50.075","Text":"of x minus 2 to the minus 2."},{"Start":"02:50.075 ","End":"02:51.980","Text":"Then we raise the power by 1."},{"Start":"02:51.980 ","End":"02:53.495","Text":"If minus 2 is n,"},{"Start":"02:53.495 ","End":"02:55.655","Text":"then n plus 1 is minus 1,"},{"Start":"02:55.655 ","End":"02:57.230","Text":"and we divide by minus 1."},{"Start":"02:57.230 ","End":"02:59.740","Text":"This is the minus 1 and this is the minus 1."},{"Start":"02:59.740 ","End":"03:01.280","Text":"Just rewriting that,"},{"Start":"03:01.280 ","End":"03:03.260","Text":"we can put the power of minus 1"},{"Start":"03:03.260 ","End":"03:05.504","Text":"as just x minus 2 in the denominator,"},{"Start":"03:05.504 ","End":"03:07.565","Text":"and the minus comes in front."},{"Start":"03:07.565 ","End":"03:10.970","Text":"Finally, I can use the property of logarithms"},{"Start":"03:10.970 ","End":"03:12.650","Text":"to say that if I have a log of something"},{"Start":"03:12.650 ","End":"03:14.285","Text":"minus the log of something,"},{"Start":"03:14.285 ","End":"03:16.745","Text":"and it\u0027s a logarithm of this over this,"},{"Start":"03:16.745 ","End":"03:18.110","Text":"which is what I\u0027ve done here."},{"Start":"03:18.110 ","End":"03:20.820","Text":"This is the answer."}],"ID":4469},{"Watched":false,"Name":"Exercise 22","Duration":"3m 3s","ChapterTopicVideoID":4461,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.190","Text":"Here we have to compute the integral of 5 minus x over x^3 plus x^2."},{"Start":"00:05.190 ","End":"00:09.600","Text":"As usual, we begin with trying to factorize the denominator."},{"Start":"00:09.600 ","End":"00:13.185","Text":"I can see that I can take x^2 out of the brackets,"},{"Start":"00:13.185 ","End":"00:17.220","Text":"and so what I get in the denominator is x^2 times x plus 1."},{"Start":"00:17.220 ","End":"00:22.590","Text":"Now it\u0027s factorized completely and we can use partial fractions."},{"Start":"00:22.590 ","End":"00:25.950","Text":"What we get is because the x is squared,"},{"Start":"00:25.950 ","End":"00:28.908","Text":"we need a representative of x and x^2,"},{"Start":"00:28.908 ","End":"00:30.743","Text":"A over this and B over this,"},{"Start":"00:30.743 ","End":"00:34.095","Text":"and C over the last factor x plus 1."},{"Start":"00:34.095 ","End":"00:38.030","Text":"We got to this from this by putting it over a common denominator."},{"Start":"00:38.030 ","End":"00:40.293","Text":"Let\u0027s see if we can do the reverse process and find out"},{"Start":"00:40.293 ","End":"00:42.830","Text":"what were the A, B, and C that led to this."},{"Start":"00:42.830 ","End":"00:47.375","Text":"Again, common denominator of x^2 times x plus 1 for each of the terms,"},{"Start":"00:47.375 ","End":"00:51.140","Text":"but then we throw out the denominator because it\u0027s the same for all the terms,"},{"Start":"00:51.140 ","End":"00:53.630","Text":"and what we get is 5 minus x,"},{"Start":"00:53.630 ","End":"00:56.162","Text":"because that\u0027s over the right common denominator,"},{"Start":"00:56.162 ","End":"00:58.745","Text":"and each of these we have to complete the missing terms."},{"Start":"00:58.745 ","End":"01:00.365","Text":"For the A, it\u0027s over x,"},{"Start":"01:00.365 ","End":"01:03.005","Text":"we\u0027re missing an x and an x plus 1, that\u0027s here."},{"Start":"01:03.005 ","End":"01:06.035","Text":"B is missing just an x plus 1, that\u0027s here."},{"Start":"01:06.035 ","End":"01:08.900","Text":"C has an x plus 1 but it\u0027s missing the x^2,"},{"Start":"01:08.900 ","End":"01:10.010","Text":"so that\u0027s that here."},{"Start":"01:10.010 ","End":"01:11.780","Text":"Now, this is not really an equation,"},{"Start":"01:11.780 ","End":"01:13.430","Text":"it\u0027s more of an identity,"},{"Start":"01:13.430 ","End":"01:15.330","Text":"meaning it\u0027s true for all x."},{"Start":"01:15.330 ","End":"01:17.000","Text":"If it\u0027s true for all x,"},{"Start":"01:17.000 ","End":"01:18.290","Text":"I can plug in what I like."},{"Start":"01:18.290 ","End":"01:20.390","Text":"For example, if I plug in x equals 0,"},{"Start":"01:20.390 ","End":"01:24.335","Text":"and I do this because I can see that that will make a lot of things 0,"},{"Start":"01:24.335 ","End":"01:26.255","Text":"the first and last become 0"},{"Start":"01:26.255 ","End":"01:29.165","Text":"and we end up with B equals 5."},{"Start":"01:29.165 ","End":"01:31.790","Text":"If I let x is equal minus 1,"},{"Start":"01:31.790 ","End":"01:34.835","Text":"I\u0027ll get rid of this and this and if you substitute your C,"},{"Start":"01:34.835 ","End":"01:39.945","Text":"we get that 6 equals 1 times C, so, of course, C equals 6."},{"Start":"01:39.945 ","End":"01:42.645","Text":"I need another value, could be anything,"},{"Start":"01:42.645 ","End":"01:44.486","Text":"take one nice small number,"},{"Start":"01:44.486 ","End":"01:47.990","Text":"and on the left, we get 5 minus 1 is 4 and etc."},{"Start":"01:47.990 ","End":"01:50.690","Text":"We get 2A, 2B, and 2C."},{"Start":"01:50.690 ","End":"01:54.080","Text":"We do know what B and C are, they\u0027re 5 and 6,"},{"Start":"01:54.080 ","End":"01:55.340","Text":"and if you plug it in,"},{"Start":"01:55.340 ","End":"01:58.475","Text":"you\u0027ll see that we end up with A equals minus 6."},{"Start":"01:58.475 ","End":"02:02.850","Text":"Now, we can put the numbers A minus 6, B is 5,"},{"Start":"02:02.850 ","End":"02:06.890","Text":"and C is 6 here and we get the integral of this, split it up,"},{"Start":"02:06.890 ","End":"02:11.705","Text":"and take the constants in front of the integral sign and we\u0027re up to this."},{"Start":"02:11.705 ","End":"02:13.337","Text":"We have a mixed bunch of things."},{"Start":"02:13.337 ","End":"02:16.316","Text":"If we have 1 over a linear term is one formula"},{"Start":"02:16.316 ","End":"02:17.825","Text":"and if it\u0027s not linear,"},{"Start":"02:17.825 ","End":"02:20.645","Text":"say, like here a quadratic, it\u0027s a different formula."},{"Start":"02:20.645 ","End":"02:23.042","Text":"At any rate, write the 1 over x^2"},{"Start":"02:23.042 ","End":"02:26.360","Text":"is x to the minus 2 because for this, I do have a formula."},{"Start":"02:26.360 ","End":"02:32.210","Text":"The linear bits, we use this and it\u0027s not 1 over x when it\u0027s x to some other power of n,"},{"Start":"02:32.210 ","End":"02:35.165","Text":"where n is not minus 1 and we use this formula."},{"Start":"02:35.165 ","End":"02:37.235","Text":"What we end up with is for the first bit,"},{"Start":"02:37.235 ","End":"02:41.780","Text":"we use this formula and we get the natural log of absolute value of x."},{"Start":"02:41.780 ","End":"02:45.905","Text":"For the last one, the integral is natural log of x plus 1."},{"Start":"02:45.905 ","End":"02:47.840","Text":"But for the middle one,"},{"Start":"02:47.840 ","End":"02:49.810","Text":"we use n equals minus 2,"},{"Start":"02:49.810 ","End":"02:51.875","Text":"n plus 1 is minus 1,"},{"Start":"02:51.875 ","End":"02:54.470","Text":"so we have a minus 1 here and a minus 1 here."},{"Start":"02:54.470 ","End":"02:58.565","Text":"If we want, we could rewrite this as putting the x on the bottom."},{"Start":"02:58.565 ","End":"03:01.574","Text":"You could write this last term as minus 5 over x,"},{"Start":"03:01.574 ","End":"03:04.560","Text":"but this will do for an answer."}],"ID":4470},{"Watched":false,"Name":"Exercise 23","Duration":"2m 53s","ChapterTopicVideoID":4462,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.230","Text":"Here we have to compute"},{"Start":"00:01.230 ","End":"00:03.810","Text":"the integral of 9x plus 36"},{"Start":"00:03.810 ","End":"00:07.815","Text":"over x cubed plus 6x squared plus 9x."},{"Start":"00:07.815 ","End":"00:11.550","Text":"As usual, we try to factorize the denominator."},{"Start":"00:11.550 ","End":"00:14.459","Text":"Certainly, I can take x outside the brackets"},{"Start":"00:14.459 ","End":"00:17.535","Text":"and I\u0027m left with x squared plus 6x plus 9."},{"Start":"00:17.535 ","End":"00:20.925","Text":"This is 1 of those perfect squares by the formula,"},{"Start":"00:20.925 ","End":"00:23.220","Text":"which is x plus 3 all squared"},{"Start":"00:23.220 ","End":"00:24.930","Text":"using the a squared plus 2ab"},{"Start":"00:24.930 ","End":"00:26.475","Text":"plus b squared formula."},{"Start":"00:26.475 ","End":"00:28.020","Text":"Now, we have it all decomposed"},{"Start":"00:28.020 ","End":"00:29.190","Text":"into linear factors,"},{"Start":"00:29.190 ","End":"00:31.020","Text":"x, x plus 3, x plus 3,"},{"Start":"00:31.020 ","End":"00:33.180","Text":"so we use partial fractions,"},{"Start":"00:33.180 ","End":"00:35.340","Text":"and we suppose that someone"},{"Start":"00:35.340 ","End":"00:37.320","Text":"had an expression of this form"},{"Start":"00:37.320 ","End":"00:39.690","Text":"and expanded it to the common"},{"Start":"00:39.690 ","End":"00:41.145","Text":"denominator to get this."},{"Start":"00:41.145 ","End":"00:42.590","Text":"Note the denominators"},{"Start":"00:42.590 ","End":"00:45.140","Text":"is a single x here, so that gives x,"},{"Start":"00:45.140 ","End":"00:46.770","Text":"x plus 3 appears squared,"},{"Start":"00:46.770 ","End":"00:49.370","Text":"so we have to take both to the power of 1"},{"Start":"00:49.370 ","End":"00:50.855","Text":"and to the power of 2."},{"Start":"00:50.855 ","End":"00:53.720","Text":"What we do is common denominator again,"},{"Start":"00:53.720 ","End":"00:55.595","Text":"and throw away the denominator."},{"Start":"00:55.595 ","End":"00:57.559","Text":"This gives us the following"},{"Start":"00:57.559 ","End":"01:00.720","Text":"after we multiply this by these 2,"},{"Start":"01:00.720 ","End":"01:02.420","Text":"and this by these 2, and so on."},{"Start":"01:02.420 ","End":"01:04.400","Text":"Now, we start plugging in values of x,"},{"Start":"01:04.400 ","End":"01:06.680","Text":"and as usual, we prefer values"},{"Start":"01:06.680 ","End":"01:08.510","Text":"of x that will make something 0."},{"Start":"01:08.510 ","End":"01:10.025","Text":"For example, if x is 0,"},{"Start":"01:10.025 ","End":"01:11.990","Text":"then there won\u0027t be a c term"},{"Start":"01:11.990 ","End":"01:15.080","Text":"and there won\u0027t be also a b term,"},{"Start":"01:15.080 ","End":"01:16.820","Text":"because we\u0027ve got x here and x here,"},{"Start":"01:16.820 ","End":"01:18.375","Text":"so that gives us 2, 0 \u0027s,"},{"Start":"01:18.375 ","End":"01:19.820","Text":"which straight away gives us"},{"Start":"01:19.820 ","End":"01:21.215","Text":"that a is equal to 4."},{"Start":"01:21.215 ","End":"01:23.360","Text":"I would try minus 3 next,"},{"Start":"01:23.360 ","End":"01:26.455","Text":"because that will make this and this 0,"},{"Start":"01:26.455 ","End":"01:28.350","Text":"so we get this,"},{"Start":"01:28.350 ","End":"01:30.165","Text":"and c is minus 3."},{"Start":"01:30.165 ","End":"01:32.270","Text":"Finally, any x will do,"},{"Start":"01:32.270 ","End":"01:33.470","Text":"for example, 1."},{"Start":"01:33.470 ","End":"01:36.740","Text":"We get another equation with A, B, and C."},{"Start":"01:36.740 ","End":"01:39.260","Text":"But we already know A and C,"},{"Start":"01:39.260 ","End":"01:42.290","Text":"A is 4 and C is minus 3."},{"Start":"01:42.290 ","End":"01:44.630","Text":"Little computation will give us,"},{"Start":"01:44.630 ","End":"01:45.755","Text":"b is minus 4."},{"Start":"01:45.755 ","End":"01:47.390","Text":"Now, that we can put in the values"},{"Start":"01:47.390 ","End":"01:49.775","Text":"for B, A, and C, the minus 4,"},{"Start":"01:49.775 ","End":"01:53.180","Text":"A is 4, and C is equal to 3,"},{"Start":"01:53.180 ","End":"01:55.625","Text":"so our integral becomes this."},{"Start":"01:55.625 ","End":"01:57.800","Text":"Let\u0027s split it up and take constants"},{"Start":"01:57.800 ","End":"02:00.680","Text":"outside of the integration sign."},{"Start":"02:00.680 ","End":"02:02.690","Text":"Here, we have these 3 integrals"},{"Start":"02:02.690 ","End":"02:04.795","Text":"with constants in front."},{"Start":"02:04.795 ","End":"02:07.700","Text":"What I want to do is these 2 are going"},{"Start":"02:07.700 ","End":"02:09.875","Text":"to work with the logarithm formula,"},{"Start":"02:09.875 ","End":"02:11.570","Text":"and this 1 is going to work"},{"Start":"02:11.570 ","End":"02:13.670","Text":"with the exponent formula."},{"Start":"02:13.670 ","End":"02:16.760","Text":"I\u0027d like to write this as the minus 2"},{"Start":"02:16.760 ","End":"02:19.225","Text":"instead of this on the denominator."},{"Start":"02:19.225 ","End":"02:22.519","Text":"We use this formula with the logarithm"},{"Start":"02:22.519 ","End":"02:24.140","Text":"for this term and this term,"},{"Start":"02:24.140 ","End":"02:26.300","Text":"and we use this formula for the last term."},{"Start":"02:26.300 ","End":"02:27.980","Text":"Simple substitutions."},{"Start":"02:27.980 ","End":"02:30.470","Text":"Here, n is equal to minus 2,"},{"Start":"02:30.470 ","End":"02:34.640","Text":"and so we get 4 natural log of x from this 1,"},{"Start":"02:34.640 ","End":"02:38.520","Text":"and minus 4 natural log of x plus 3 for this 1."},{"Start":"02:38.520 ","End":"02:40.695","Text":"For this, we raise the power by 1."},{"Start":"02:40.695 ","End":"02:44.100","Text":"The n plus 1 is minus 1 over minus 1."},{"Start":"02:44.100 ","End":"02:45.410","Text":"I personally would write"},{"Start":"02:45.410 ","End":"02:48.770","Text":"the last term as minus minus 3,"},{"Start":"02:48.770 ","End":"02:51.050","Text":"which is 3 over x plus 3,"},{"Start":"02:51.050 ","End":"02:52.800","Text":"but it\u0027s fine as it is."},{"Start":"02:52.800 ","End":"02:54.490","Text":"We\u0027re done."}],"ID":4471},{"Watched":false,"Name":"Exercise 24","Duration":"3m 32s","ChapterTopicVideoID":4463,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.100","Text":"Here, we have to compute the integral"},{"Start":"00:02.100 ","End":"00:04.830","Text":"of a rational function 1 over this"},{"Start":"00:04.830 ","End":"00:06.840","Text":"expression times this expression."},{"Start":"00:06.840 ","End":"00:09.930","Text":"As usual, we try and factorize the denominator."},{"Start":"00:09.930 ","End":"00:12.450","Text":"Well, it is factorized, but not completely."},{"Start":"00:12.450 ","End":"00:14.535","Text":"This should look familiar."},{"Start":"00:14.535 ","End":"00:18.600","Text":"After all, this is just the perfect square of x minus 1,"},{"Start":"00:18.600 ","End":"00:20.490","Text":"and here x minus 2."},{"Start":"00:20.490 ","End":"00:22.320","Text":"What I\u0027m saying is that,"},{"Start":"00:22.320 ","End":"00:24.600","Text":"this is x minus 1 squared,"},{"Start":"00:24.600 ","End":"00:26.430","Text":"the famous x squared minus 2x, plus 1,"},{"Start":"00:26.430 ","End":"00:28.140","Text":"and this is x minus 2 squared."},{"Start":"00:28.140 ","End":"00:29.790","Text":"You should recognize this right away."},{"Start":"00:29.790 ","End":"00:30.930","Text":"But if not, you could always"},{"Start":"00:30.930 ","End":"00:32.610","Text":"solve the quadratic equation"},{"Start":"00:32.610 ","End":"00:34.740","Text":"and get that there\u0027s only 1 root here,"},{"Start":"00:34.740 ","End":"00:36.795","Text":"and that\u0027s 2, and so on."},{"Start":"00:36.795 ","End":"00:38.870","Text":"Now, that we have this,"},{"Start":"00:38.870 ","End":"00:41.510","Text":"we can do a partial fraction decomposition."},{"Start":"00:41.510 ","End":"00:44.630","Text":"This is squared, so it appears both"},{"Start":"00:44.630 ","End":"00:47.030","Text":"as x minus 1 and x minus 1 squared."},{"Start":"00:47.030 ","End":"00:48.830","Text":"Similarly, we have an x minus 2"},{"Start":"00:48.830 ","End":"00:50.480","Text":"and an x minus 2 squared"},{"Start":"00:50.480 ","End":"00:52.400","Text":"with 4 different constants."},{"Start":"00:52.400 ","End":"00:54.320","Text":"In order to find out what were"},{"Start":"00:54.320 ","End":"00:56.090","Text":"the A, B, and C, and D,"},{"Start":"00:56.090 ","End":"00:57.305","Text":"that led to this,"},{"Start":"00:57.305 ","End":"00:59.180","Text":"let\u0027s put a common denominator,"},{"Start":"00:59.180 ","End":"01:01.580","Text":"which will be this whole thing here."},{"Start":"01:01.580 ","End":"01:03.170","Text":"Then after we throw away"},{"Start":"01:03.170 ","End":"01:04.550","Text":"the common denominator,"},{"Start":"01:04.550 ","End":"01:07.010","Text":"what we\u0027ll get is the 1 from here,"},{"Start":"01:07.010 ","End":"01:08.930","Text":"and then each constant in the numerator,"},{"Start":"01:08.930 ","End":"01:11.750","Text":"we multiply by whatever factors it\u0027s missing."},{"Start":"01:11.750 ","End":"01:13.655","Text":"Here, we\u0027re missing an x minus 1,"},{"Start":"01:13.655 ","End":"01:15.170","Text":"and an x minus 2 squared."},{"Start":"01:15.170 ","End":"01:17.330","Text":"Here we\u0027re just missing x minus 2 squared,"},{"Start":"01:17.330 ","End":"01:20.465","Text":"and so on, and we get this expression."},{"Start":"01:20.465 ","End":"01:22.250","Text":"This is not really an equality,"},{"Start":"01:22.250 ","End":"01:23.960","Text":"it\u0027s actually an identity."},{"Start":"01:23.960 ","End":"01:25.460","Text":"It holds true for all x."},{"Start":"01:25.460 ","End":"01:27.140","Text":"We substitute whatever x"},{"Start":"01:27.140 ","End":"01:28.565","Text":"is convenient for us."},{"Start":"01:28.565 ","End":"01:29.930","Text":"I would certainly want to try"},{"Start":"01:29.930 ","End":"01:32.180","Text":"x equals 1 and x equals 2,"},{"Start":"01:32.180 ","End":"01:34.385","Text":"because that would make certain things 0."},{"Start":"01:34.385 ","End":"01:36.935","Text":"Let\u0027s start off with x equals 1,"},{"Start":"01:36.935 ","End":"01:38.840","Text":"and this is what we get."},{"Start":"01:38.840 ","End":"01:42.185","Text":"All of them are 0 except for the B term,"},{"Start":"01:42.185 ","End":"01:44.300","Text":"so that gives us that B is 1."},{"Start":"01:44.300 ","End":"01:46.850","Text":"Then we can try x equals 2,"},{"Start":"01:46.850 ","End":"01:49.670","Text":"and then also 3 terms will become 0,"},{"Start":"01:49.670 ","End":"01:51.065","Text":"those with x minus 2."},{"Start":"01:51.065 ","End":"01:51.800","Text":"Then from this,"},{"Start":"01:51.800 ","End":"01:53.615","Text":"we can get what D is,"},{"Start":"01:53.615 ","End":"01:56.570","Text":"and then we try any values of x,"},{"Start":"01:56.570 ","End":"01:58.940","Text":"say x equals 0."},{"Start":"01:58.940 ","End":"02:01.745","Text":"Now, we have 4 variables,"},{"Start":"02:01.745 ","End":"02:04.490","Text":"but we do have the value of B"},{"Start":"02:04.490 ","End":"02:05.540","Text":"and the value of D,"},{"Start":"02:05.540 ","End":"02:07.370","Text":"so that makes things a bit better."},{"Start":"02:07.370 ","End":"02:09.320","Text":"What we\u0027re left with after"},{"Start":"02:09.320 ","End":"02:11.255","Text":"we move sides and everything,"},{"Start":"02:11.255 ","End":"02:12.530","Text":"we have an expression"},{"Start":"02:12.530 ","End":"02:14.320","Text":"in 2 variables, A and C."},{"Start":"02:14.320 ","End":"02:17.105","Text":"Similarly, if I let x equal 3,"},{"Start":"02:17.105 ","End":"02:18.800","Text":"then I also get something"},{"Start":"02:18.800 ","End":"02:20.090","Text":"where B and D are known,"},{"Start":"02:20.090 ","End":"02:22.130","Text":"but I still get something in A and C."},{"Start":"02:22.130 ","End":"02:25.445","Text":"I have to solve 2 equations in 2 unknowns,"},{"Start":"02:25.445 ","End":"02:29.060","Text":"A and C, and these are the 2 equations."},{"Start":"02:29.060 ","End":"02:32.215","Text":"In the end, we get that A is 2,"},{"Start":"02:32.215 ","End":"02:33.560","Text":"and C is minus 2."},{"Start":"02:33.560 ","End":"02:35.960","Text":"You know how to do this thing."},{"Start":"02:35.960 ","End":"02:37.685","Text":"We have all the constants now,"},{"Start":"02:37.685 ","End":"02:39.380","Text":"A, C, B, and D."},{"Start":"02:39.380 ","End":"02:43.140","Text":"We can plug them in to this part here,"},{"Start":"02:43.140 ","End":"02:46.640","Text":"and what we will get is the integral of this."},{"Start":"02:46.640 ","End":"02:48.200","Text":"Then we just separate it"},{"Start":"02:48.200 ","End":"02:49.850","Text":"into 4 separate integrals"},{"Start":"02:49.850 ","End":"02:51.440","Text":"and take constants out."},{"Start":"02:51.440 ","End":"02:53.690","Text":"We take out 2 from here, 1 from here,"},{"Start":"02:53.690 ","End":"02:55.130","Text":"minus 2 and 1,"},{"Start":"02:55.130 ","End":"02:56.825","Text":"and we end up with this."},{"Start":"02:56.825 ","End":"02:58.424","Text":"There\u0027s 2 kinds."},{"Start":"02:58.424 ","End":"03:01.270","Text":"There\u0027s the 1 with 1 over a linear,"},{"Start":"03:01.270 ","End":"03:03.530","Text":"that will give us a natural log,"},{"Start":"03:03.530 ","End":"03:05.390","Text":"and an x plus a constant"},{"Start":"03:05.390 ","End":"03:07.550","Text":"to a power other than minus 1,"},{"Start":"03:07.550 ","End":"03:09.080","Text":"we\u0027ll use a different formula."},{"Start":"03:09.080 ","End":"03:10.370","Text":"This is the formula,"},{"Start":"03:10.370 ","End":"03:11.780","Text":"1 over x plus a,"},{"Start":"03:11.780 ","End":"03:13.820","Text":"for here and for here."},{"Start":"03:13.820 ","End":"03:16.040","Text":"The x plus a to the power of n,"},{"Start":"03:16.040 ","End":"03:17.580","Text":"we\u0027ll use here and here."},{"Start":"03:17.580 ","End":"03:19.130","Text":"If we substitute all that,"},{"Start":"03:19.130 ","End":"03:20.825","Text":"this is what we\u0027ll get."},{"Start":"03:20.825 ","End":"03:22.880","Text":"I would suggest simplifying"},{"Start":"03:22.880 ","End":"03:24.500","Text":"still further by bringing this"},{"Start":"03:24.500 ","End":"03:26.660","Text":"x minus 2 onto the denominator"},{"Start":"03:26.660 ","End":"03:28.760","Text":"and taking the minus 1s in front,"},{"Start":"03:28.760 ","End":"03:29.945","Text":"and changing the sign."},{"Start":"03:29.945 ","End":"03:31.520","Text":"But this is fine as is."},{"Start":"03:31.520 ","End":"03:33.570","Text":"We are done."}],"ID":4472},{"Watched":false,"Name":"Exercise 25","Duration":"3m 22s","ChapterTopicVideoID":4464,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"Here, we have to compute the integral of x plus 4 over x minus 1 cubed."},{"Start":"00:06.030 ","End":"00:11.880","Text":"Well, that\u0027s nice because the denominator is already completely factorized for us."},{"Start":"00:11.880 ","End":"00:15.630","Text":"We can go ahead and use partial fractions."},{"Start":"00:15.630 ","End":"00:19.200","Text":"Now, notice that this term is to the power of 3."},{"Start":"00:19.200 ","End":"00:20.940","Text":"When I do partial fractions,"},{"Start":"00:20.940 ","End":"00:25.125","Text":"I have to have a representative of all the powers up to 3."},{"Start":"00:25.125 ","End":"00:27.765","Text":"We get an x minus 1,"},{"Start":"00:27.765 ","End":"00:29.535","Text":"an x minus 1 squared,"},{"Start":"00:29.535 ","End":"00:33.660","Text":"and an x minus 1 cubed with constants over it."},{"Start":"00:33.660 ","End":"00:36.525","Text":"To find these constants,"},{"Start":"00:36.525 ","End":"00:39.900","Text":"we put everything over a common denominator,"},{"Start":"00:39.900 ","End":"00:42.145","Text":"which is x minus 1 cubed,"},{"Start":"00:42.145 ","End":"00:45.735","Text":"and then just compare the numerators."},{"Start":"00:45.735 ","End":"00:48.530","Text":"What we get is the x plus 4 from here,"},{"Start":"00:48.530 ","End":"00:51.230","Text":"A times whatever was missing,"},{"Start":"00:51.230 ","End":"00:52.670","Text":"which is x minus 1 squared."},{"Start":"00:52.670 ","End":"00:54.230","Text":"B was missing an x minus 1,"},{"Start":"00:54.230 ","End":"00:56.695","Text":"and C is also over the same denominator."},{"Start":"00:56.695 ","End":"00:58.560","Text":"This is what we get,"},{"Start":"00:58.560 ","End":"01:00.590","Text":"and now we substitute values."},{"Start":"01:00.590 ","End":"01:03.710","Text":"The first thing I would do is put x equals 1,"},{"Start":"01:03.710 ","End":"01:09.710","Text":"because that will make both this 0 and I\u0027m left with what c is,"},{"Start":"01:09.710 ","End":"01:11.240","Text":"which is equal to 5,"},{"Start":"01:11.240 ","End":"01:13.220","Text":"and then any values of x,"},{"Start":"01:13.220 ","End":"01:16.715","Text":"for example, x equals 2."},{"Start":"01:16.715 ","End":"01:20.225","Text":"If we substitute, this is what we get."},{"Start":"01:20.225 ","End":"01:22.835","Text":"I\u0027m not going into all the details."},{"Start":"01:22.835 ","End":"01:25.670","Text":"We know that c is 5,"},{"Start":"01:25.670 ","End":"01:29.480","Text":"and so what we get is that"},{"Start":"01:29.480 ","End":"01:35.270","Text":"a plus b is equal to 1 after we take the 5 over to the other side."},{"Start":"01:35.270 ","End":"01:37.475","Text":"If we try another value of x,"},{"Start":"01:37.475 ","End":"01:40.280","Text":"0 is good to try because it\u0027s easy usually."},{"Start":"01:40.280 ","End":"01:45.925","Text":"We get this and we get another equation in A and B,"},{"Start":"01:45.925 ","End":"01:49.025","Text":"2 equations and n 2 unknowns."},{"Start":"01:49.025 ","End":"01:52.070","Text":"It\u0027s easy enough even mentally to do."},{"Start":"01:52.070 ","End":"01:53.630","Text":"If I add the 2 equations,"},{"Start":"01:53.630 ","End":"01:56.870","Text":"I get that 2A is 0, so A is 0."},{"Start":"01:56.870 ","End":"01:59.245","Text":"I subtract the 2 equations,"},{"Start":"01:59.245 ","End":"02:01.410","Text":"I get that 2B is 2,"},{"Start":"02:01.410 ","End":"02:03.420","Text":"so B is 1."},{"Start":"02:03.420 ","End":"02:06.210","Text":"The C was always equal to 5,"},{"Start":"02:06.210 ","End":"02:08.645","Text":"so we have these and now we can put them here,"},{"Start":"02:08.645 ","End":"02:10.265","Text":"here, and here."},{"Start":"02:10.265 ","End":"02:13.995","Text":"Our integral becomes this."},{"Start":"02:13.995 ","End":"02:16.424","Text":"Of course, we can throw out the one with the 0,"},{"Start":"02:16.424 ","End":"02:18.900","Text":"split this into 2 pieces,"},{"Start":"02:18.900 ","End":"02:21.555","Text":"and take the 5 out of the integral."},{"Start":"02:21.555 ","End":"02:25.300","Text":"We have 1 of these and 5 of these,"},{"Start":"02:25.300 ","End":"02:28.130","Text":"and this denominator is to the power of 2,"},{"Start":"02:28.130 ","End":"02:29.465","Text":"this to the power of 3."},{"Start":"02:29.465 ","End":"02:35.105","Text":"We want to use the formula for x minus 1^n,"},{"Start":"02:35.105 ","End":"02:38.520","Text":"x minus 1 or x, same thing."},{"Start":"02:38.520 ","End":"02:45.875","Text":"In each case, let\u0027s just write it as to the power of minus 2 so we can use the formula,"},{"Start":"02:45.875 ","End":"02:48.740","Text":"and then what we do is we raise the power by 1 and"},{"Start":"02:48.740 ","End":"02:51.800","Text":"divide by the new power, the new exponent."},{"Start":"02:51.800 ","End":"02:54.830","Text":"Here, we get minus 2 plus 1 is minus 1,"},{"Start":"02:54.830 ","End":"02:57.815","Text":"so this is to the minus 1 over minus 1."},{"Start":"02:57.815 ","End":"03:00.360","Text":"Here, we add 1 to get minus 2,"},{"Start":"03:00.360 ","End":"03:03.360","Text":"so minus 2 here and minus 2 here."},{"Start":"03:03.360 ","End":"03:09.575","Text":"Finally, just rearranging it or put the power of minus 1 back to the denominator here,"},{"Start":"03:09.575 ","End":"03:12.875","Text":"instead of the minus, put it in the denominator."},{"Start":"03:12.875 ","End":"03:20.030","Text":"When the minus out front and the minus here from here and so on,"},{"Start":"03:20.030 ","End":"03:22.080","Text":"and this is the answer."}],"ID":4473},{"Watched":false,"Name":"Exercise 26","Duration":"2m 1s","ChapterTopicVideoID":4465,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"Here we have to compute this integral,"},{"Start":"00:02.160 ","End":"00:05.985","Text":"6x squared minus 4x plus 1 over x minus 1 cubed."},{"Start":"00:05.985 ","End":"00:09.135","Text":"The denominator is already completely factorized,"},{"Start":"00:09.135 ","End":"00:14.475","Text":"so we can jump straight to the partial fraction bit and because it\u0027s to the power of 3,"},{"Start":"00:14.475 ","End":"00:15.900","Text":"we need x minus 1,"},{"Start":"00:15.900 ","End":"00:17.535","Text":"we need x minus 1 squared,"},{"Start":"00:17.535 ","End":"00:20.610","Text":"and we need x minus 1 cubed as here,"},{"Start":"00:20.610 ","End":"00:22.560","Text":"each with a constant over it."},{"Start":"00:22.560 ","End":"00:26.100","Text":"Now, we put a common denominator and we get that this is equal"},{"Start":"00:26.100 ","End":"00:29.925","Text":"to a times x minus 1 squared and so on and so on."},{"Start":"00:29.925 ","End":"00:32.595","Text":"We get this equation which is really an identity,"},{"Start":"00:32.595 ","End":"00:35.550","Text":"which means that we can plug in different values of x."},{"Start":"00:35.550 ","End":"00:39.180","Text":"We need to put 3 values of x in because we have 3 unknowns."},{"Start":"00:39.180 ","End":"00:41.145","Text":"First thing I would put x equals 1,"},{"Start":"00:41.145 ","End":"00:43.185","Text":"because that makes both of these 0,"},{"Start":"00:43.185 ","End":"00:45.150","Text":"so we\u0027re left with the value of C,"},{"Start":"00:45.150 ","End":"00:47.130","Text":"and then any 2 values, for example,"},{"Start":"00:47.130 ","End":"00:49.595","Text":"x equals 2 will give us this equation."},{"Start":"00:49.595 ","End":"00:54.905","Text":"We already know C, so we get an equation in A and B, which is this."},{"Start":"00:54.905 ","End":"00:56.510","Text":"If we try another value of x,"},{"Start":"00:56.510 ","End":"00:58.175","Text":"say 0, we\u0027ll get this."},{"Start":"00:58.175 ","End":"01:02.960","Text":"Again, we have an equation in a and b because we know what C is."},{"Start":"01:02.960 ","End":"01:05.420","Text":"Here\u0027s 2 equations in A and B,"},{"Start":"01:05.420 ","End":"01:07.595","Text":"and we know how to solve this thing."},{"Start":"01:07.595 ","End":"01:10.145","Text":"Turns out that A is 6, and B is 8,"},{"Start":"01:10.145 ","End":"01:11.360","Text":"and C was 3,"},{"Start":"01:11.360 ","End":"01:12.715","Text":"we already found this out."},{"Start":"01:12.715 ","End":"01:15.360","Text":"We can put these values in for A, B,"},{"Start":"01:15.360 ","End":"01:18.800","Text":"and C here, and what we get is that our integral is this."},{"Start":"01:18.800 ","End":"01:22.460","Text":"Let\u0027s split it up take constants outside the integral sign,"},{"Start":"01:22.460 ","End":"01:24.035","Text":"and so we have this."},{"Start":"01:24.035 ","End":"01:25.685","Text":"Looking at the denominators,"},{"Start":"01:25.685 ","End":"01:28.310","Text":"I have an x minus 1 in the denominator,"},{"Start":"01:28.310 ","End":"01:31.280","Text":"this is going to be using the natural logarithm formula,"},{"Start":"01:31.280 ","End":"01:33.800","Text":"but these 2 will be using a different formula for"},{"Start":"01:33.800 ","End":"01:36.380","Text":"the power of n. The ones on the denominator,"},{"Start":"01:36.380 ","End":"01:37.910","Text":"I\u0027ll write as negative exponents,"},{"Start":"01:37.910 ","End":"01:39.950","Text":"so it\u0027ll be easier to use the formula."},{"Start":"01:39.950 ","End":"01:43.745","Text":"This part gives us 6 times natural logarithm of the denominator."},{"Start":"01:43.745 ","End":"01:45.950","Text":"This will use the exponent minus 2,"},{"Start":"01:45.950 ","End":"01:47.360","Text":"we raise it by 1,"},{"Start":"01:47.360 ","End":"01:50.240","Text":"we get minus 1 and we divide by that minus 1."},{"Start":"01:50.240 ","End":"01:55.205","Text":"Similarly here, minus 3 raised by 1 is minus 2 over minus 2."},{"Start":"01:55.205 ","End":"02:02.160","Text":"Then this bit of rearranging minuses in front and we have this as our answer."}],"ID":4474},{"Watched":false,"Name":"Exercise 27","Duration":"1m 28s","ChapterTopicVideoID":4466,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"Here, we have to compute the integral of x plus 4 over x minus 1 cubed."},{"Start":"00:04.830 ","End":"00:07.695","Text":"Now, normally we would use partial fractions."},{"Start":"00:07.695 ","End":"00:10.605","Text":"This is completely decomposed in the denominator,"},{"Start":"00:10.605 ","End":"00:12.450","Text":"but there are other techniques as well,"},{"Start":"00:12.450 ","End":"00:15.175","Text":"and 1 of them is substitution."},{"Start":"00:15.175 ","End":"00:16.920","Text":"Just for the sake of variety,"},{"Start":"00:16.920 ","End":"00:18.555","Text":"we\u0027ll use substitution here."},{"Start":"00:18.555 ","End":"00:22.160","Text":"What I\u0027m going to do is I\u0027m going to let t equals x minus 1,"},{"Start":"00:22.160 ","End":"00:24.440","Text":"and then dx is equal to dt."},{"Start":"00:24.440 ","End":"00:29.355","Text":"After the substitution, I get x plus 4 over this x minus 1 is t,"},{"Start":"00:29.355 ","End":"00:31.215","Text":"so it\u0027s cubed and dt."},{"Start":"00:31.215 ","End":"00:35.090","Text":"But this is still not good because we still have x and not all in terms of t,"},{"Start":"00:35.090 ","End":"00:38.390","Text":"but it\u0027s easy to see that x is equal to t plus 1."},{"Start":"00:38.390 ","End":"00:40.055","Text":"We have a reverse substitution."},{"Start":"00:40.055 ","End":"00:43.490","Text":"So letting x equals t plus 1, we get this,"},{"Start":"00:43.490 ","End":"00:46.175","Text":"and just collecting terms on the numerator,"},{"Start":"00:46.175 ","End":"00:49.060","Text":"we get t plus 5 over t cubed."},{"Start":"00:49.060 ","End":"00:51.440","Text":"Now, I can split this into 2 pieces,"},{"Start":"00:51.440 ","End":"00:54.395","Text":"t over t cubed and 5 over t cubed,"},{"Start":"00:54.395 ","End":"00:58.430","Text":"and this gives us t over t cubed is 1 over t squared,"},{"Start":"00:58.430 ","End":"01:00.080","Text":"which is t^minus 2,"},{"Start":"01:00.080 ","End":"01:01.850","Text":"and here we have t^minus 3."},{"Start":"01:01.850 ","End":"01:05.075","Text":"So we have a sum of 2 exponents in t,"},{"Start":"01:05.075 ","End":"01:07.160","Text":"and we have a formula for exponents;"},{"Start":"01:07.160 ","End":"01:09.110","Text":"whenever the exponent is not minus 1,"},{"Start":"01:09.110 ","End":"01:12.755","Text":"we just raise the exponent by 1 and divide by it, that will give us,"},{"Start":"01:12.755 ","End":"01:14.750","Text":"here, a minus 1 over minus 1,"},{"Start":"01:14.750 ","End":"01:17.540","Text":"and here, a minus 2 over a minus 2."},{"Start":"01:17.540 ","End":"01:21.290","Text":"You have to remember, after a substitution to substitute back."},{"Start":"01:21.290 ","End":"01:24.650","Text":"Remember that t was x minus 1, so back here,"},{"Start":"01:24.650 ","End":"01:29.460","Text":"we can put t as x minus 1 and this gives us our final answer."}],"ID":4475},{"Watched":false,"Name":"Exercise 28","Duration":"1m 25s","ChapterTopicVideoID":4467,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.665","Text":"We\u0027ve had this exercise before and we solved it by means of substitution."},{"Start":"00:04.665 ","End":"00:08.250","Text":"Now I\u0027m going to do it using a different technique and of course,"},{"Start":"00:08.250 ","End":"00:10.500","Text":"I can still use partial fractions to solve"},{"Start":"00:10.500 ","End":"00:13.095","Text":"this since the denominator is completely decomposed."},{"Start":"00:13.095 ","End":"00:16.410","Text":"But I want to show you yet another method without substitution and without"},{"Start":"00:16.410 ","End":"00:20.820","Text":"partial fractions just by messing around algebraically, let\u0027s say."},{"Start":"00:20.820 ","End":"00:23.505","Text":"I see I have an x minus 1 here,"},{"Start":"00:23.505 ","End":"00:26.670","Text":"and it would be nice if it was x minus 1 over here too."},{"Start":"00:26.670 ","End":"00:31.950","Text":"What I do is I make it x minus 1 and then compensate by putting plus 5."},{"Start":"00:31.950 ","End":"00:33.330","Text":"What good does this do me?"},{"Start":"00:33.330 ","End":"00:35.175","Text":"I could split it up into 2 bits."},{"Start":"00:35.175 ","End":"00:38.780","Text":"I can put the x minus 1 separately over x minus 1 cubed,"},{"Start":"00:38.780 ","End":"00:43.510","Text":"and the 5 separately or here I can cancel and I can reduce this 3 to a 2,"},{"Start":"00:43.510 ","End":"00:46.070","Text":"and after this, I can just separate into"},{"Start":"00:46.070 ","End":"00:49.430","Text":"2 integrals and take constants front of the integral sign."},{"Start":"00:49.430 ","End":"00:52.100","Text":"So I end up with 1 over x minus 1 squared."},{"Start":"00:52.100 ","End":"00:55.340","Text":"I\u0027ll just write it in terms of negative exponents and this 5,"},{"Start":"00:55.340 ","End":"00:59.535","Text":"I also write the cubed on the denominator as a power of minus 3."},{"Start":"00:59.535 ","End":"01:02.900","Text":"Now I can just integrate it because x minus 1 is"},{"Start":"01:02.900 ","End":"01:06.380","Text":"practically the same as x because it\u0027s got derivative 1 also."},{"Start":"01:06.380 ","End":"01:09.320","Text":"I can just imagine it was x to the minus 2"},{"Start":"01:09.320 ","End":"01:12.290","Text":"and then I raise this minus 2 by 1 and divide by it."},{"Start":"01:12.290 ","End":"01:17.990","Text":"Similarly, raise the minus 3 by 1 to minus 2 and divide by it and just tidy up a bit,"},{"Start":"01:17.990 ","End":"01:20.540","Text":"we can put these things in the denominator and make"},{"Start":"01:20.540 ","End":"01:23.480","Text":"the exponents positive and the minus comes on top,"},{"Start":"01:23.480 ","End":"01:26.010","Text":"and so on and we are done."}],"ID":4476},{"Watched":false,"Name":"Exercise 29","Duration":"1m 45s","ChapterTopicVideoID":4468,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.955","Text":"In this exercise, we have to compute the integral of"},{"Start":"00:02.955 ","End":"00:07.485","Text":"2x plus 3 over x squared minus 2x plus 1^4."},{"Start":"00:07.485 ","End":"00:09.480","Text":"Now, if you look at the denominator,"},{"Start":"00:09.480 ","End":"00:13.335","Text":"this x squared minus 2x plus 1 is a perfect square."},{"Start":"00:13.335 ","End":"00:15.480","Text":"It\u0027s x minus 1 all squared."},{"Start":"00:15.480 ","End":"00:19.170","Text":"So that means that I can rewrite it like this and then"},{"Start":"00:19.170 ","End":"00:22.890","Text":"combine the 2 and the 4 to make it to the power of 8."},{"Start":"00:22.890 ","End":"00:26.810","Text":"Denominator is completely decomposed or factorized but I would"},{"Start":"00:26.810 ","End":"00:31.100","Text":"not like to do it with partial fractions because then I\u0027d have to have x minus 1,"},{"Start":"00:31.100 ","End":"00:32.740","Text":"x minus 1 squared, cubed,"},{"Start":"00:32.740 ","End":"00:35.465","Text":"4th and so on up to the power of 8, a real mess."},{"Start":"00:35.465 ","End":"00:38.285","Text":"Instead, let\u0027s use the method of substitution."},{"Start":"00:38.285 ","End":"00:41.270","Text":"What I\u0027m going to do is let x minus 1 equal t and"},{"Start":"00:41.270 ","End":"00:44.480","Text":"then dx is equal to dt and after I substitute,"},{"Start":"00:44.480 ","End":"00:46.430","Text":"I get x minus 1 is t^8."},{"Start":"00:46.430 ","End":"00:52.010","Text":"You can also substitute x in terms of t. If I just put the minus 1 on the other side,"},{"Start":"00:52.010 ","End":"00:53.510","Text":"I get x is t plus 1,"},{"Start":"00:53.510 ","End":"00:54.740","Text":"and that\u0027s what I\u0027ve done here."},{"Start":"00:54.740 ","End":"00:56.495","Text":"A little bit of simplification,"},{"Start":"00:56.495 ","End":"01:00.890","Text":"2t plus 2 plus 3 is just 2t plus 5 over t^8."},{"Start":"01:00.890 ","End":"01:02.960","Text":"Now, I\u0027m going to split this up into 2 bits."},{"Start":"01:02.960 ","End":"01:07.490","Text":"2t over t^8 is separate and 5 over t^8 is separate,"},{"Start":"01:07.490 ","End":"01:12.670","Text":"but then, this becomes 2 times t^ minus 7, t^1 minus 8."},{"Start":"01:12.670 ","End":"01:14.175","Text":"1 minus 8 is minus 7,"},{"Start":"01:14.175 ","End":"01:16.670","Text":"and here just t^ minus 8."},{"Start":"01:16.670 ","End":"01:19.715","Text":"Then just by using the formula for exponents,"},{"Start":"01:19.715 ","End":"01:22.850","Text":"raise the power by 1 and divide by it."},{"Start":"01:22.850 ","End":"01:25.385","Text":"So we get minus 6 over minus 6 here,"},{"Start":"01:25.385 ","End":"01:27.830","Text":"minus 7 over minus 7."},{"Start":"01:27.830 ","End":"01:31.860","Text":"Then I have to remember to substitute back, t was x minus 1,"},{"Start":"01:31.860 ","End":"01:34.775","Text":"so we put that, and finally,"},{"Start":"01:34.775 ","End":"01:38.600","Text":"we can just play with it a bit to make it nicer set of a negative exponent,"},{"Start":"01:38.600 ","End":"01:40.280","Text":"I can put it on the denominator,"},{"Start":"01:40.280 ","End":"01:42.529","Text":"the minuses I can bring up front,"},{"Start":"01:42.529 ","End":"01:44.780","Text":"and this is what we\u0027re left with."},{"Start":"01:44.780 ","End":"01:46.200","Text":"We\u0027re done."}],"ID":4477},{"Watched":false,"Name":"Exercise 30","Duration":"2m 29s","ChapterTopicVideoID":4469,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"Here we have to compute the following integral,"},{"Start":"00:02.340 ","End":"00:06.240","Text":"2x squared plus 2x plus 1 over x squared plus 1 times x plus 2."},{"Start":"00:06.240 ","End":"00:08.910","Text":"Notice that the denominator has degree 3,"},{"Start":"00:08.910 ","End":"00:12.450","Text":"2 plus 1 is 3, and the numerator has degree 2, so that\u0027s fine."},{"Start":"00:12.450 ","End":"00:13.995","Text":"This is less than this in degree."},{"Start":"00:13.995 ","End":"00:16.920","Text":"Also, the denominators is completely factorized."},{"Start":"00:16.920 ","End":"00:20.130","Text":"This is linear and this is an irreducible quadratic,"},{"Start":"00:20.130 ","End":"00:22.290","Text":"x squared plus 1 can\u0027t be equal to 0,"},{"Start":"00:22.290 ","End":"00:24.255","Text":"so it has no roots, so it\u0027s irreducible."},{"Start":"00:24.255 ","End":"00:28.950","Text":"All that remains is the partial fractions part of the exercise."},{"Start":"00:28.950 ","End":"00:31.730","Text":"For each irreducible quadratic like this,"},{"Start":"00:31.730 ","End":"00:33.350","Text":"we put an Ax plus B,"},{"Start":"00:33.350 ","End":"00:35.150","Text":"a linear term, and for each linear,"},{"Start":"00:35.150 ","End":"00:38.375","Text":"we put a constant C. This has got to equal this."},{"Start":"00:38.375 ","End":"00:40.610","Text":"We imagine that someone started out with something"},{"Start":"00:40.610 ","End":"00:43.010","Text":"like this and used a common denominator to get here,"},{"Start":"00:43.010 ","End":"00:44.180","Text":"we\u0027ll do the reverse."},{"Start":"00:44.180 ","End":"00:46.970","Text":"Again, we\u0027ll put a common denominator of x squared plus 1,"},{"Start":"00:46.970 ","End":"00:49.745","Text":"x plus 2 and compare numerators."},{"Start":"00:49.745 ","End":"00:53.825","Text":"What we get is the 2x squared plus 2x plus 1 is equal to this,"},{"Start":"00:53.825 ","End":"00:55.210","Text":"but it\u0027s not just equal,"},{"Start":"00:55.210 ","End":"00:56.689","Text":"it\u0027s actually an identity,"},{"Start":"00:56.689 ","End":"00:58.445","Text":"meaning it\u0027s true for all x."},{"Start":"00:58.445 ","End":"01:01.265","Text":"If it\u0027s true for all x, we can put in whatever x we want."},{"Start":"01:01.265 ","End":"01:05.915","Text":"A good idea might be negative 2 because then this part would be 0,"},{"Start":"01:05.915 ","End":"01:09.260","Text":"and so we most easily find 1 of the constant C,"},{"Start":"01:09.260 ","End":"01:11.390","Text":"and C was equal to 1,"},{"Start":"01:11.390 ","End":"01:13.550","Text":"then we have to try another value of x."},{"Start":"01:13.550 ","End":"01:15.680","Text":"We can\u0027t make this 0 or anything else 0,"},{"Start":"01:15.680 ","End":"01:19.250","Text":"so we often try 0 because it\u0027s easy to compute."},{"Start":"01:19.250 ","End":"01:23.480","Text":"We get this equation which leads to B equals 0, and similarly,"},{"Start":"01:23.480 ","End":"01:25.325","Text":"put another value of x, say 1,"},{"Start":"01:25.325 ","End":"01:28.820","Text":"this is the equation we get and this leads to A equals 1."},{"Start":"01:28.820 ","End":"01:32.240","Text":"Now that we have all the constants A, B, and C,"},{"Start":"01:32.240 ","End":"01:33.905","Text":"we can put them in here,"},{"Start":"01:33.905 ","End":"01:35.990","Text":"here, and here, and finally,"},{"Start":"01:35.990 ","End":"01:39.980","Text":"rewrite this in terms of partial fractions because B is 0,"},{"Start":"01:39.980 ","End":"01:41.420","Text":"so this part doesn\u0027t appear,"},{"Start":"01:41.420 ","End":"01:43.250","Text":"so it\u0027s just A and C as one."},{"Start":"01:43.250 ","End":"01:46.310","Text":"Now, we have to treat each of these pieces separately."},{"Start":"01:46.310 ","End":"01:48.770","Text":"Whenever we have 1 over a linear thing,"},{"Start":"01:48.770 ","End":"01:50.885","Text":"there\u0027s a rule with the logarithm."},{"Start":"01:50.885 ","End":"01:55.010","Text":"If we had the denominator with derivative on the numerator,"},{"Start":"01:55.010 ","End":"01:58.430","Text":"that\u0027s another trick, and we almost have that. In fact, we could fix it."},{"Start":"01:58.430 ","End":"01:59.450","Text":"If I put 2 here,"},{"Start":"01:59.450 ","End":"02:02.195","Text":"we would have the derivative of the denominator."},{"Start":"02:02.195 ","End":"02:06.470","Text":"We put the 2 there but we correct it by putting a 2 here also,"},{"Start":"02:06.470 ","End":"02:09.640","Text":"and this one is still just as is."},{"Start":"02:09.640 ","End":"02:13.805","Text":"Finally, using the logarithm of the denominator here, well,"},{"Start":"02:13.805 ","End":"02:16.820","Text":"there is a formula that f prime over f has an"},{"Start":"02:16.820 ","End":"02:20.480","Text":"integral of natural log of absolute value of f. So I apply it twice,"},{"Start":"02:20.480 ","End":"02:22.340","Text":"once here and once here,"},{"Start":"02:22.340 ","End":"02:23.795","Text":"the 1/2 stays of course,"},{"Start":"02:23.795 ","End":"02:26.165","Text":"so we get natural log of the denominator,"},{"Start":"02:26.165 ","End":"02:30.540","Text":"and here also, plus the constant of integration, and we\u0027re done."}],"ID":4478},{"Watched":false,"Name":"Exercise 31","Duration":"1m 51s","ChapterTopicVideoID":4470,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.650","Text":"Here we have to compute the integral"},{"Start":"00:01.650 ","End":"00:03.960","Text":"of 2x squared plus x minus 1"},{"Start":"00:03.960 ","End":"00:06.450","Text":"over x squared plus 1x minus 3."},{"Start":"00:06.450 ","End":"00:08.670","Text":"Note that the numerator has degree 2,"},{"Start":"00:08.670 ","End":"00:11.040","Text":"denominator has degree 3, 2 plus 1,"},{"Start":"00:11.040 ","End":"00:12.840","Text":"and so that\u0027s all okay."},{"Start":"00:12.840 ","End":"00:14.340","Text":"This degree is less than here,"},{"Start":"00:14.340 ","End":"00:16.110","Text":"and we would normally factorize"},{"Start":"00:16.110 ","End":"00:17.280","Text":"the denominator but someone\u0027s"},{"Start":"00:17.280 ","End":"00:18.990","Text":"done all the work for us already,"},{"Start":"00:18.990 ","End":"00:20.910","Text":"and we have it factorized to linear"},{"Start":"00:20.910 ","End":"00:23.070","Text":"and irreducible quadratic."},{"Start":"00:23.070 ","End":"00:26.640","Text":"We\u0027ll use partial fractions and write it as;"},{"Start":"00:26.640 ","End":"00:28.050","Text":"for the irreducible quadratic,"},{"Start":"00:28.050 ","End":"00:29.160","Text":"we need a linear term."},{"Start":"00:29.160 ","End":"00:31.634","Text":"For the linear, we need just a constant."},{"Start":"00:31.634 ","End":"00:33.390","Text":"Multiplying out,"},{"Start":"00:33.390 ","End":"00:34.260","Text":"putting everything over"},{"Start":"00:34.260 ","End":"00:35.310","Text":"a common denominator,"},{"Start":"00:35.310 ","End":"00:37.065","Text":"and then throwing out the denominator."},{"Start":"00:37.065 ","End":"00:39.529","Text":"What we end up with is this equation,"},{"Start":"00:39.529 ","End":"00:41.240","Text":"which is really an identity,"},{"Start":"00:41.240 ","End":"00:42.875","Text":"meaning it\u0027s true for all x."},{"Start":"00:42.875 ","End":"00:45.530","Text":"If I put for example x equals 3"},{"Start":"00:45.530 ","End":"00:46.775","Text":"to make this 0,"},{"Start":"00:46.775 ","End":"00:50.510","Text":"we will get 20 is 0 plus 10C,"},{"Start":"00:50.510 ","End":"00:52.910","Text":"and therefore we get that C is 2."},{"Start":"00:52.910 ","End":"00:54.030","Text":"We have 3 constants,"},{"Start":"00:54.030 ","End":"00:56.765","Text":"so we need 2 more values of x to put in."},{"Start":"00:56.765 ","End":"00:58.340","Text":"If we put in x equals 0,"},{"Start":"00:58.340 ","End":"01:00.170","Text":"this is what we get an equation"},{"Start":"01:00.170 ","End":"01:01.070","Text":"with B and C,"},{"Start":"01:01.070 ","End":"01:02.405","Text":"but we know C already,"},{"Start":"01:02.405 ","End":"01:04.785","Text":"so this gives us that B is 1."},{"Start":"01:04.785 ","End":"01:06.240","Text":"Finally another value,"},{"Start":"01:06.240 ","End":"01:08.570","Text":"say x equals 1, I often use,"},{"Start":"01:08.570 ","End":"01:10.780","Text":"and we get this with A, B, and C."},{"Start":"01:10.780 ","End":"01:12.130","Text":"But we know B and C."},{"Start":"01:12.130 ","End":"01:13.960","Text":"All we have is to find A,"},{"Start":"01:13.960 ","End":"01:16.040","Text":"and it comes out that A is 0."},{"Start":"01:16.040 ","End":"01:17.375","Text":"I put those here,"},{"Start":"01:17.375 ","End":"01:19.250","Text":"notice that the Ax term disappears,"},{"Start":"01:19.250 ","End":"01:21.065","Text":"so we just have constants."},{"Start":"01:21.065 ","End":"01:23.720","Text":"This constant stays and this constant stays."},{"Start":"01:23.720 ","End":"01:26.765","Text":"What we get is the integral of 1,"},{"Start":"01:26.765 ","End":"01:28.500","Text":"the B over x squared plus 1,"},{"Start":"01:28.500 ","End":"01:31.745","Text":"and 2 from C over x minus 3,"},{"Start":"01:31.745 ","End":"01:33.020","Text":"split that up into 2 first"},{"Start":"01:33.020 ","End":"01:34.400","Text":"and take constants out."},{"Start":"01:34.400 ","End":"01:35.690","Text":"This 1 is an immediate"},{"Start":"01:35.690 ","End":"01:37.790","Text":"integral of the arctangent."},{"Start":"01:37.790 ","End":"01:40.160","Text":"This 1 is a familiar 1,"},{"Start":"01:40.160 ","End":"01:42.320","Text":"the derivative of the denominator"},{"Start":"01:42.320 ","End":"01:43.895","Text":"is on the numerator."},{"Start":"01:43.895 ","End":"01:45.859","Text":"The answer is natural logarithm"},{"Start":"01:45.859 ","End":"01:47.929","Text":"of the absolute value of the denominator."},{"Start":"01:47.929 ","End":"01:50.360","Text":"Finally, we add the constant of integration."},{"Start":"01:50.360 ","End":"01:51.960","Text":"We\u0027re done."}],"ID":4479},{"Watched":false,"Name":"Exercise 32","Duration":"2m 43s","ChapterTopicVideoID":4471,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.445","Text":"Here we have to compute the integral of 3 over x squared plus 1 x squared plus 4."},{"Start":"00:05.445 ","End":"00:08.505","Text":"Note that both of these are irreducible quadratics"},{"Start":"00:08.505 ","End":"00:11.130","Text":"so the denominators are completely factorized,"},{"Start":"00:11.130 ","End":"00:14.205","Text":"decomposed, and we\u0027re going to use partial fractions."},{"Start":"00:14.205 ","End":"00:16.080","Text":"Now for each irreducible quadratic,"},{"Start":"00:16.080 ","End":"00:17.355","Text":"we get a linear term."},{"Start":"00:17.355 ","End":"00:18.480","Text":"For the x squared plus 1,"},{"Start":"00:18.480 ","End":"00:19.680","Text":"we get 1 linear thing,"},{"Start":"00:19.680 ","End":"00:23.640","Text":"Ax plus B and for the x squared plus 4 we\u0027ll put Cx plus D. If"},{"Start":"00:23.640 ","End":"00:27.630","Text":"we make a common denominator and compare the numerators,"},{"Start":"00:27.630 ","End":"00:29.205","Text":"what we\u0027ll get is this."},{"Start":"00:29.205 ","End":"00:31.650","Text":"Now, normally what we do is we treat this as"},{"Start":"00:31.650 ","End":"00:34.470","Text":"an identity and we substitute different values of x."},{"Start":"00:34.470 ","End":"00:36.840","Text":"We\u0027d have to substitute 4 different values of x to"},{"Start":"00:36.840 ","End":"00:39.600","Text":"get 4 equations and 4 unknowns, A, B, C,"},{"Start":"00:39.600 ","End":"00:43.280","Text":"and D. There is a slightly more efficient method and that"},{"Start":"00:43.280 ","End":"00:47.195","Text":"is to completely expand this and then we get a polynomial."},{"Start":"00:47.195 ","End":"00:50.420","Text":"We compare the coefficients of x to the power of 1,"},{"Start":"00:50.420 ","End":"00:53.120","Text":"x to the power of 2, x to the power of 3,"},{"Start":"00:53.120 ","End":"00:54.415","Text":"and the free co-efficient."},{"Start":"00:54.415 ","End":"00:56.195","Text":"I\u0027ll show you. Let\u0027s expand this."},{"Start":"00:56.195 ","End":"01:00.245","Text":"After this, we just collect together the terms of x^0,"},{"Start":"01:00.245 ","End":"01:02.330","Text":"x^1, x squared and x cubed all separately."},{"Start":"01:02.330 ","End":"01:04.490","Text":"For example, if we want x^1,"},{"Start":"01:04.490 ","End":"01:09.545","Text":"then we see that we have a 4A and C. If we wanted x cubed,"},{"Start":"01:09.545 ","End":"01:11.930","Text":"we would take it from here and get a C,"},{"Start":"01:11.930 ","End":"01:13.100","Text":"and from here we\u0027d get an A,"},{"Start":"01:13.100 ","End":"01:16.250","Text":"so it\u0027s A plus C. Collect together the terms and then we can"},{"Start":"01:16.250 ","End":"01:19.580","Text":"compare the coefficients of x^0."},{"Start":"01:19.580 ","End":"01:24.510","Text":"Then we\u0027ll compare the coefficient of x to the power of 1 and x^2x and the 3."},{"Start":"01:24.510 ","End":"01:26.225","Text":"Let\u0027s see what we did here."},{"Start":"01:26.225 ","End":"01:30.890","Text":"This coefficient of x^0, 4B plus D has got to equal 3."},{"Start":"01:30.890 ","End":"01:32.375","Text":"That\u0027s the free co-efficient."},{"Start":"01:32.375 ","End":"01:34.250","Text":"It\u0027s like a 1 there, x^0."},{"Start":"01:34.250 ","End":"01:37.370","Text":"Then if we look here 4A plus C is the coefficient of x,"},{"Start":"01:37.370 ","End":"01:39.830","Text":"there is no coefficient of x here, so that\u0027s 0."},{"Start":"01:39.830 ","End":"01:42.680","Text":"Similarly with the x squared coefficient is got to be"},{"Start":"01:42.680 ","End":"01:46.370","Text":"0 and the x cubed coefficient is also going to be 0."},{"Start":"01:46.370 ","End":"01:48.740","Text":"This is 4 equations and 4 unknowns,"},{"Start":"01:48.740 ","End":"01:49.970","Text":"but they don\u0027t all appear."},{"Start":"01:49.970 ","End":"01:53.915","Text":"For example, we can group them and here we have 2 equations in B and D,"},{"Start":"01:53.915 ","End":"01:57.635","Text":"and here 2 equations in A and C. That\u0027s much easier to solve than"},{"Start":"01:57.635 ","End":"02:01.625","Text":"4 equations and 4 unknowns or separately, 2 equations and 2 unknowns,"},{"Start":"02:01.625 ","End":"02:03.230","Text":"D is 1, D is minus 1,"},{"Start":"02:03.230 ","End":"02:05.225","Text":"A is 0, C is 0."},{"Start":"02:05.225 ","End":"02:08.300","Text":"What we want to do now is plug them into here."},{"Start":"02:08.300 ","End":"02:10.385","Text":"What we get is this,"},{"Start":"02:10.385 ","End":"02:11.720","Text":"because of the zeros,"},{"Start":"02:11.720 ","End":"02:14.930","Text":"the x terms, B and D are both 0,"},{"Start":"02:14.930 ","End":"02:18.080","Text":"so what we\u0027re left with is a 1 and a minus 1 and"},{"Start":"02:18.080 ","End":"02:21.590","Text":"splitting them up and putting coefficients in front, we get this."},{"Start":"02:21.590 ","End":"02:24.585","Text":"This is a very standard integral."},{"Start":"02:24.585 ","End":"02:27.035","Text":"It\u0027s the arctangent of x here,"},{"Start":"02:27.035 ","End":"02:33.505","Text":"but here it\u0027s 1 half because it\u0027s 1 over the square root of 4 arctangent of x over 2."},{"Start":"02:33.505 ","End":"02:36.140","Text":"Here is the formula. When it\u0027s a instead of 1,"},{"Start":"02:36.140 ","End":"02:37.955","Text":"then we just put the square root of a,"},{"Start":"02:37.955 ","End":"02:39.185","Text":"square root of 4 is 2."},{"Start":"02:39.185 ","End":"02:44.240","Text":"Over here, and over here we have another 2. That\u0027s the answer."}],"ID":4480},{"Watched":false,"Name":"Exercise 33","Duration":"3m 33s","ChapterTopicVideoID":4472,"CourseChapterTopicPlaylistID":3686,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.590","Text":"In this exercise, we have to find"},{"Start":"00:01.590 ","End":"00:04.110","Text":"the integral of 1/x times"},{"Start":"00:04.110 ","End":"00:05.985","Text":"x squared plus 1 squared."},{"Start":"00:05.985 ","End":"00:07.710","Text":"The denominator is completely"},{"Start":"00:07.710 ","End":"00:10.110","Text":"factorized into a linear factor,"},{"Start":"00:10.110 ","End":"00:12.599","Text":"and the square of an irreducible quadratic,"},{"Start":"00:12.599 ","End":"00:14.355","Text":"x squared plus 1 is irreducible."},{"Start":"00:14.355 ","End":"00:15.480","Text":"There are many ways of doing this."},{"Start":"00:15.480 ","End":"00:18.255","Text":"Let\u0027s try the partial fractions method."},{"Start":"00:18.255 ","End":"00:20.130","Text":"If we use partial fractions,"},{"Start":"00:20.130 ","End":"00:22.390","Text":"then what we do is decompose it."},{"Start":"00:22.390 ","End":"00:24.810","Text":"As for the x, it\u0027s only to the power of 1,"},{"Start":"00:24.810 ","End":"00:27.150","Text":"so we just have A/x,"},{"Start":"00:27.150 ","End":"00:29.930","Text":"x squared plus 1 appears to the power of 2,"},{"Start":"00:29.930 ","End":"00:32.595","Text":"so we need a representative for each power,"},{"Start":"00:32.595 ","End":"00:34.350","Text":"for the power 1 and power 2,"},{"Start":"00:34.350 ","End":"00:35.130","Text":"and in each case,"},{"Start":"00:35.130 ","End":"00:37.055","Text":"we put a linear factor on the top."},{"Start":"00:37.055 ","End":"00:38.060","Text":"This is what we get."},{"Start":"00:38.060 ","End":"00:39.320","Text":"If we multiply out"},{"Start":"00:39.320 ","End":"00:40.790","Text":"and put a common denominator,"},{"Start":"00:40.790 ","End":"00:42.920","Text":"we get the following equation,"},{"Start":"00:42.920 ","End":"00:44.600","Text":"which is really an identity."},{"Start":"00:44.600 ","End":"00:47.210","Text":"We have 5 unknown quantities;"},{"Start":"00:47.210 ","End":"00:49.150","Text":"A, B, C, D, and E,"},{"Start":"00:49.150 ","End":"00:51.800","Text":"1 way to do it is to substitute different"},{"Start":"00:51.800 ","End":"00:54.150","Text":"values of x and get 5 equations"},{"Start":"00:54.150 ","End":"00:55.335","Text":"and 5 unknowns,"},{"Start":"00:55.335 ","End":"00:56.640","Text":"which is a real mess."},{"Start":"00:56.640 ","End":"00:57.930","Text":"There is another way."},{"Start":"00:57.930 ","End":"00:59.904","Text":"I\u0027ve done it before in an exercise,"},{"Start":"00:59.904 ","End":"01:02.180","Text":"where we just expand this completely"},{"Start":"01:02.180 ","End":"01:04.790","Text":"into a polynomial and compare terms"},{"Start":"01:04.790 ","End":"01:06.080","Text":"with the same degree."},{"Start":"01:06.080 ","End":"01:09.230","Text":"If I open up the brackets, this is what I get."},{"Start":"01:09.230 ","End":"01:11.505","Text":"Then I collect like terms."},{"Start":"01:11.505 ","End":"01:12.940","Text":"What we get is,"},{"Start":"01:12.940 ","End":"01:15.430","Text":"if we compare the free coefficients,"},{"Start":"01:15.430 ","End":"01:17.920","Text":"here we have 1 and here we have A."},{"Start":"01:17.920 ","End":"01:20.520","Text":"If we compare coefficients of x,"},{"Start":"01:20.520 ","End":"01:23.360","Text":"we find that there\u0027s an x here and here,"},{"Start":"01:23.360 ","End":"01:25.730","Text":"so that gives us C plus E, and so on,"},{"Start":"01:25.730 ","End":"01:28.790","Text":"up to x^4th, which we have here and here,"},{"Start":"01:28.790 ","End":"01:30.065","Text":"so that\u0027s A plus B."},{"Start":"01:30.065 ","End":"01:31.909","Text":"Now, each of the powers"},{"Start":"01:31.909 ","End":"01:33.800","Text":"has to appear with the same coefficient"},{"Start":"01:33.800 ","End":"01:35.870","Text":"on both sides of the identity."},{"Start":"01:35.870 ","End":"01:39.080","Text":"If we take x^0 or free coefficients,"},{"Start":"01:39.080 ","End":"01:41.195","Text":"we can say that 1 equals A."},{"Start":"01:41.195 ","End":"01:44.420","Text":"If we take the coefficient of x^1,"},{"Start":"01:44.420 ","End":"01:46.190","Text":"we get that there\u0027s nothing here."},{"Start":"01:46.190 ","End":"01:47.120","Text":"All the other coefficients"},{"Start":"01:47.120 ","End":"01:49.180","Text":"are going to be 0, is C plus E,"},{"Start":"01:49.180 ","End":"01:51.365","Text":"and this is also going to be 0,"},{"Start":"01:51.365 ","End":"01:52.670","Text":"the x squared coefficients."},{"Start":"01:52.670 ","End":"01:54.290","Text":"For the x cubed coefficient,"},{"Start":"01:54.290 ","End":"01:55.820","Text":"we\u0027re going to get C is 0,"},{"Start":"01:55.820 ","End":"01:57.410","Text":"and for the x^4th coefficient,"},{"Start":"01:57.410 ","End":"01:59.420","Text":"we\u0027ll get that A plus B is 0."},{"Start":"01:59.420 ","End":"02:00.410","Text":"This is the answer,"},{"Start":"02:00.410 ","End":"02:01.970","Text":"A is 1, B is minus 1."},{"Start":"02:01.970 ","End":"02:03.440","Text":"C, 0; E, 0;"},{"Start":"02:03.440 ","End":"02:04.895","Text":"and D is minus 1."},{"Start":"02:04.895 ","End":"02:07.265","Text":"Now, we can put these values up here,"},{"Start":"02:07.265 ","End":"02:09.560","Text":"noticing that C and E are 0,"},{"Start":"02:09.560 ","End":"02:11.930","Text":"so we don\u0027t have any free coefficient"},{"Start":"02:11.930 ","End":"02:13.250","Text":"over the quadratics."},{"Start":"02:13.250 ","End":"02:15.380","Text":"In fact, what we get is just by"},{"Start":"02:15.380 ","End":"02:18.430","Text":"substituting these values in here,"},{"Start":"02:18.430 ","End":"02:21.605","Text":"we can break it up into 3 separate integrals"},{"Start":"02:21.605 ","End":"02:25.130","Text":"and we can work on each of these 3 separately."},{"Start":"02:25.130 ","End":"02:26.300","Text":"That\u0027s called an asterisk,"},{"Start":"02:26.300 ","End":"02:28.700","Text":"double asterisk, and triple asterisk."},{"Start":"02:28.700 ","End":"02:30.080","Text":"We\u0027re going to use the formula"},{"Start":"02:30.080 ","End":"02:32.210","Text":"that if the numerator is the derivative"},{"Start":"02:32.210 ","End":"02:33.079","Text":"of the denominator,"},{"Start":"02:33.079 ","End":"02:35.480","Text":"the answer is natural log of the denominator."},{"Start":"02:35.480 ","End":"02:37.430","Text":"Another rule is that,"},{"Start":"02:37.430 ","End":"02:39.470","Text":"if we have f squared at the bottom"},{"Start":"02:39.470 ","End":"02:40.730","Text":"and f prime at the top,"},{"Start":"02:40.730 ","End":"02:42.830","Text":"then the answer is minus 1/f."},{"Start":"02:42.830 ","End":"02:43.760","Text":"Easy to see if you just"},{"Start":"02:43.760 ","End":"02:45.320","Text":"differentiate this anyway."},{"Start":"02:45.320 ","End":"02:47.380","Text":"The first 1 is the integral of 1/x,"},{"Start":"02:47.380 ","End":"02:48.830","Text":"and this we know very well"},{"Start":"02:48.830 ","End":"02:50.210","Text":"to be natural log of x."},{"Start":"02:50.210 ","End":"02:51.635","Text":"We don\u0027t even have to use this."},{"Start":"02:51.635 ","End":"02:55.310","Text":"The 1 with 2 asterisks was x/x squared plus 1,"},{"Start":"02:55.310 ","End":"02:57.500","Text":"but our usual trick of putting a 2 here"},{"Start":"02:57.500 ","End":"02:59.090","Text":"and then compensating here,"},{"Start":"02:59.090 ","End":"03:00.860","Text":"will leave us to 1/2 times"},{"Start":"03:00.860 ","End":"03:02.885","Text":"the natural log of the denominator,"},{"Start":"03:02.885 ","End":"03:04.790","Text":"and the 1 with 3 asterisks,"},{"Start":"03:04.790 ","End":"03:06.050","Text":"again instead of x,"},{"Start":"03:06.050 ","End":"03:08.475","Text":"we put it as 1/2 times 2x."},{"Start":"03:08.475 ","End":"03:10.790","Text":"We actually have this formula,"},{"Start":"03:10.790 ","End":"03:12.905","Text":"because if x squared plus 1 is f,"},{"Start":"03:12.905 ","End":"03:14.450","Text":"then here we have f squared"},{"Start":"03:14.450 ","End":"03:16.310","Text":"and here we have f prime at the top,"},{"Start":"03:16.310 ","End":"03:19.325","Text":"so the answer\u0027s just minus 1/f."},{"Start":"03:19.325 ","End":"03:21.500","Text":"But don\u0027t forget the 1/2."},{"Start":"03:21.500 ","End":"03:24.020","Text":"Now, getting back to these 3 exercises"},{"Start":"03:24.020 ","End":"03:25.970","Text":"with the asterisks on the other page."},{"Start":"03:25.970 ","End":"03:28.370","Text":"I\u0027m just copying the answers from there."},{"Start":"03:28.370 ","End":"03:29.510","Text":"I\u0027m putting them into here,"},{"Start":"03:29.510 ","End":"03:30.800","Text":"and adding the constant"},{"Start":"03:30.800 ","End":"03:32.330","Text":"of integration at the end."},{"Start":"03:32.330 ","End":"03:34.410","Text":"We\u0027re done here."}],"ID":4481}],"Thumbnail":null,"ID":3686}]

[{"ID":3686,"Videos":[8497,4488,8498,4491,4490,4492,4493,4494,4495,4496,4497,4449,4450,4451,4452,4453,4454,4455,4456,4457,4458,4459,4460,4461,4462,4463,4464,4465,4466,4467,4468,4469,4470,4471,4472,4473,4474,4475,4476,4477,4478,4479,4480,4481]}];

[8497,4488,8498,4491,4490,4492,4493,4494,4495,4496,4497,4449,4450,4451,4452,4453,4454,4455,4456,4457,4458,4459,4460,4461,4462,4463,4464,4465,4466,4467,4468,4469,4470,4471,4472,4473,4474,4475,4476,4477,4478,4479,4480,4481];

1.1

1

Get unlimited access to **1500 subjects** including **personalised modules**

Start your free trial
We couldn't find any results for

Upload your syllabus now and our team will create a customised module especially for you!

Alert

and we will create a personalised module (just for you) in less than **48 hours...**