Introduction to Integration by Parts
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- Integration by Parts
- Integration by Parts (continued)
- 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

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[{"Name":"Introduction to Integration by Parts","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Integration by Parts","Duration":"15m 21s","ChapterTopicVideoID":4428,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/4428.jpeg","UploadDate":"2019-12-11T21:01:34.0530000","DurationForVideoObject":"PT15M21S","Description":null,"MetaTitle":"Integration by Parts: Video + Workbook | Proprep","MetaDescription":"Integration by Parts - Introduction to Integration by Parts. 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-by-parts/introduction-to-integration-by-parts/vid4412","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.920","Text":"In this clip, I\u0027ll be talking about integration by parts and before I start,"},{"Start":"00:04.920 ","End":"00:10.335","Text":"I\u0027d like to do some warm-up exercises which will help us. I\u0027ll give an example."},{"Start":"00:10.335 ","End":"00:16.890","Text":"If I say that u is equal to e to the power of x,"},{"Start":"00:16.890 ","End":"00:20.264","Text":"and I say, what does du equal?"},{"Start":"00:20.264 ","End":"00:21.885","Text":"That\u0027s a question mark."},{"Start":"00:21.885 ","End":"00:25.770","Text":"Then the answer is we just differentiate this with respect to x,"},{"Start":"00:25.770 ","End":"00:29.400","Text":"which is also e to the x and we put a dx here."},{"Start":"00:29.400 ","End":"00:32.325","Text":"If u equals x,"},{"Start":"00:32.325 ","End":"00:34.830","Text":"what does du equal to?"},{"Start":"00:34.830 ","End":"00:37.110","Text":"du is equal to 1dx."},{"Start":"00:37.110 ","End":"00:41.860","Text":"In other words, du is just equal to dx as if we stuck a d in front of both."},{"Start":"00:41.860 ","End":"00:49.085","Text":"The third example, if u is equal to the natural logarithm of x, what is du?"},{"Start":"00:49.085 ","End":"00:51.800","Text":"Well, it\u0027s just the derivative of this, dx."},{"Start":"00:51.800 ","End":"00:55.450","Text":"In other words, 1 over x dx."},{"Start":"00:55.450 ","End":"00:57.330","Text":"Okay. So far, so good."},{"Start":"00:57.330 ","End":"01:00.000","Text":"Let\u0027s try a reverse exercise."},{"Start":"01:00.000 ","End":"01:02.400","Text":"Suppose I tell you that,"},{"Start":"01:02.400 ","End":"01:04.335","Text":"and I\u0027ll switch from u to v,"},{"Start":"01:04.335 ","End":"01:07.895","Text":"that dv and let\u0027s not look at this."},{"Start":"01:07.895 ","End":"01:14.465","Text":"You know what, I\u0027m going to scroll up even so you don\u0027t see what I\u0027ve written there."},{"Start":"01:14.465 ","End":"01:24.005","Text":"If I say that dv is e to the x dx and I ask what v equals?"},{"Start":"01:24.005 ","End":"01:26.840","Text":"Then we would say that to get from dv to v,"},{"Start":"01:26.840 ","End":"01:31.235","Text":"we do an integration and the integral of e to the x"},{"Start":"01:31.235 ","End":"01:37.300","Text":"is just e to the x and if I say that dv equals dx,"},{"Start":"01:37.300 ","End":"01:43.335","Text":"then I can look at it as 1dx and v is the integral of 1 which is just x."},{"Start":"01:43.335 ","End":"01:51.400","Text":"My third example, if dv were equal to 1 over x dx,"},{"Start":"01:51.400 ","End":"01:52.965","Text":"what would v equal?"},{"Start":"01:52.965 ","End":"02:00.520","Text":"Then it\u0027s just the integral of 1 over x and possible answer could be natural log of x."},{"Start":"02:00.520 ","End":"02:04.560","Text":"Could have a constant, no need to, and that\u0027s it."},{"Start":"02:04.560 ","End":"02:09.050","Text":"This is exactly the exercise we\u0027re going to need when we do"},{"Start":"02:09.050 ","End":"02:13.585","Text":"integration by parts and let me get into that now."},{"Start":"02:13.585 ","End":"02:17.390","Text":"First I\u0027d like to just clear the board and"},{"Start":"02:17.390 ","End":"02:21.005","Text":"now I\u0027d like to write the formula for integration by parts."},{"Start":"02:21.005 ","End":"02:29.630","Text":"What this formula says is that the integral of udv is equal"},{"Start":"02:29.630 ","End":"02:38.510","Text":"to uv minus the integral of vdu."},{"Start":"02:38.510 ","End":"02:45.109","Text":"This formula tries to help us to deal with a situation where we have a product."},{"Start":"02:45.109 ","End":"02:54.230","Text":"For example, I might have the integral of x times e to the x dx,"},{"Start":"02:54.230 ","End":"03:03.950","Text":"in which case, this might be u and this might be dv and this is the product."},{"Start":"03:03.950 ","End":"03:06.125","Text":"That\u0027s the first thing I noticed."},{"Start":"03:06.125 ","End":"03:10.475","Text":"The second thing is that we have an integral here,"},{"Start":"03:10.475 ","End":"03:13.670","Text":"but we see that what\u0027s on the right hand side also"},{"Start":"03:13.670 ","End":"03:17.000","Text":"contains an integral so you might say what\u0027s the point?"},{"Start":"03:17.000 ","End":"03:18.770","Text":"We had an integral and now again,"},{"Start":"03:18.770 ","End":"03:20.659","Text":"we have to compute an integral."},{"Start":"03:20.659 ","End":"03:22.580","Text":"Well, the answer is very simple."},{"Start":"03:22.580 ","End":"03:28.355","Text":"Hopefully, this integral is easier to do than this integral,"},{"Start":"03:28.355 ","End":"03:31.625","Text":"or maybe even this is impossible and this is very possible."},{"Start":"03:31.625 ","End":"03:36.440","Text":"Let\u0027s continue with this example and see how this will be useful."},{"Start":"03:36.440 ","End":"03:39.170","Text":"According to the formula,"},{"Start":"03:39.170 ","End":"03:43.065","Text":"this is going to equal uv,"},{"Start":"03:43.065 ","End":"03:45.180","Text":"let me just put placeholders,"},{"Start":"03:45.180 ","End":"03:46.910","Text":"I\u0027m going to put a u here,"},{"Start":"03:46.910 ","End":"03:50.045","Text":"I\u0027m going to put v here,"},{"Start":"03:50.045 ","End":"03:53.960","Text":"and then a minus sign and an integral,"},{"Start":"03:53.960 ","End":"03:57.380","Text":"and then here I\u0027m going to put v,"},{"Start":"03:57.380 ","End":"04:00.770","Text":"and here I\u0027m going to put du."},{"Start":"04:00.770 ","End":"04:02.435","Text":"Let\u0027s see how we do this."},{"Start":"04:02.435 ","End":"04:09.450","Text":"Well, u we already have that\u0027s x and now we come to the warm-up exercises."},{"Start":"04:09.450 ","End":"04:13.535","Text":"If dv is e to the x dx,"},{"Start":"04:13.535 ","End":"04:16.705","Text":"then v is just the integral of this,"},{"Start":"04:16.705 ","End":"04:18.975","Text":"which is e to the x."},{"Start":"04:18.975 ","End":"04:21.585","Text":"That was 1 of the warm-up exercises."},{"Start":"04:21.585 ","End":"04:24.730","Text":"That\u0027s v and once again here we have v,"},{"Start":"04:24.730 ","End":"04:26.965","Text":"so I just copy that e to the x."},{"Start":"04:26.965 ","End":"04:28.885","Text":"Now I have du here,"},{"Start":"04:28.885 ","End":"04:30.250","Text":"but I have u."},{"Start":"04:30.250 ","End":"04:32.380","Text":"Once again, in the warm-up exercises,"},{"Start":"04:32.380 ","End":"04:34.510","Text":"we showed that if u is equal to x,"},{"Start":"04:34.510 ","End":"04:37.910","Text":"then du is dx."},{"Start":"04:37.940 ","End":"04:40.480","Text":"If I get rid of all this,"},{"Start":"04:40.480 ","End":"04:41.905","Text":"well, I\u0027ll keep it for the moment,"},{"Start":"04:41.905 ","End":"04:46.465","Text":"what I\u0027ve got is I have to compute this integral as follows."},{"Start":"04:46.465 ","End":"04:54.055","Text":"Let it equal xe to the x minus the integral of e to the x dx."},{"Start":"04:54.055 ","End":"04:56.630","Text":"Like I said, we had an integral and we still have an"},{"Start":"04:56.630 ","End":"05:00.245","Text":"integral but isn\u0027t this integral much easier to compute?"},{"Start":"05:00.245 ","End":"05:03.060","Text":"After all, we know the integral of e to the x,"},{"Start":"05:03.060 ","End":"05:06.155","Text":"so our answer will be just copying xe to the x."},{"Start":"05:06.155 ","End":"05:09.710","Text":"This integral is e to the x also and finally,"},{"Start":"05:09.710 ","End":"05:12.040","Text":"plus the constant of integration."},{"Start":"05:12.040 ","End":"05:15.845","Text":"This is our first example of how to use integration by parts."},{"Start":"05:15.845 ","End":"05:20.599","Text":"The word parts is because they are factors in a product that\u0027s the parts."},{"Start":"05:20.599 ","End":"05:24.260","Text":"Let\u0027s examine more closely what\u0027s happening here."},{"Start":"05:24.260 ","End":"05:27.005","Text":"Essentially, if I ignore the dx,"},{"Start":"05:27.005 ","End":"05:28.790","Text":"I have a product of 2 functions."},{"Start":"05:28.790 ","End":"05:34.884","Text":"I have x and I have e to the x and 1 of the functions x,"},{"Start":"05:34.884 ","End":"05:41.870","Text":"I differentiated because from x I get to 1."},{"Start":"05:41.870 ","End":"05:48.389","Text":"The other function, I integrated from e to the x,"},{"Start":"05:48.389 ","End":"05:52.065","Text":"I got e to the x,"},{"Start":"05:52.065 ","End":"05:53.625","Text":"but as an integral."},{"Start":"05:53.625 ","End":"05:59.010","Text":"You see, in 1 case when I go from u to du,"},{"Start":"05:59.510 ","End":"06:06.865","Text":"that\u0027s differentiating and when I go from dv to v,"},{"Start":"06:06.865 ","End":"06:09.165","Text":"I should really put arrows here,"},{"Start":"06:09.165 ","End":"06:13.610","Text":"and in this, yeah, when I go here,"},{"Start":"06:13.610 ","End":"06:20.715","Text":"I differentiate from u to du and from dv to v I integrate."},{"Start":"06:20.715 ","End":"06:23.665","Text":"That\u0027s basically what happens here."},{"Start":"06:23.665 ","End":"06:28.255","Text":"We have a product of 2 functions, we differentiate 1,"},{"Start":"06:28.255 ","End":"06:29.770","Text":"we integrate the other,"},{"Start":"06:29.770 ","End":"06:35.080","Text":"we get a new product and hopefully this is easier to compute as an integral."},{"Start":"06:35.080 ","End":"06:38.455","Text":"I just copied this exercise again because I want to show you something."},{"Start":"06:38.455 ","End":"06:41.500","Text":"I actually had another choice for a product."},{"Start":"06:41.500 ","End":"06:44.500","Text":"Theoretically, I could have said, okay,"},{"Start":"06:44.500 ","End":"06:51.580","Text":"let this 1 be u and x dx could have been dv."},{"Start":"06:51.770 ","End":"06:58.605","Text":"Whereas here, I integrated e to the x and differentiated x,"},{"Start":"06:58.605 ","End":"07:00.680","Text":"what would have happened is I would have"},{"Start":"07:00.680 ","End":"07:05.210","Text":"integrated x and got at somewhere along the process,"},{"Start":"07:05.210 ","End":"07:08.990","Text":"x squared over 2 and I would have differentiated the e to"},{"Start":"07:08.990 ","End":"07:12.845","Text":"the x and I would have ended up getting this integral."},{"Start":"07:12.845 ","End":"07:15.875","Text":"I would have gotten the uv minus,"},{"Start":"07:15.875 ","End":"07:18.125","Text":"and this would be the vdu,"},{"Start":"07:18.125 ","End":"07:23.300","Text":"and so this was actually worse than the original exercise,"},{"Start":"07:23.300 ","End":"07:25.160","Text":"whether I put it this way,"},{"Start":"07:25.160 ","End":"07:28.670","Text":"that\u0027s the correct choice and get an easier integral."},{"Start":"07:28.670 ","End":"07:31.760","Text":"Sometimes, if we do it the wrong way,"},{"Start":"07:31.760 ","End":"07:35.150","Text":"we get an integral that\u0027s impossible and our hand is forced,"},{"Start":"07:35.150 ","End":"07:37.330","Text":"as we\u0027ll see in the next example."},{"Start":"07:37.330 ","End":"07:38.840","Text":"Okay. Clear the board."},{"Start":"07:38.840 ","End":"07:43.460","Text":"The next example is going to be the integral"},{"Start":"07:43.460 ","End":"07:50.835","Text":"of x to the 10th natural log of x dx."},{"Start":"07:50.835 ","End":"07:53.650","Text":"I erase the arrows."},{"Start":"07:53.650 ","End":"08:01.465","Text":"Just want to remind you that when we go from u to du, we differentiate."},{"Start":"08:01.465 ","End":"08:06.025","Text":"When we go from dv to v,"},{"Start":"08:06.025 ","End":"08:11.995","Text":"we integrate, differentiate, integrate."},{"Start":"08:11.995 ","End":"08:14.350","Text":"We have to decide when we have a product"},{"Start":"08:14.350 ","End":"08:16.420","Text":"which you want to differentiate, which to integrate."},{"Start":"08:16.420 ","End":"08:19.585","Text":"Now here we have a product x to the 10th of natural log of x."},{"Start":"08:19.585 ","End":"08:24.040","Text":"Either integrate this and differentiate this or the other way around."},{"Start":"08:24.040 ","End":"08:26.650","Text":"Now it\u0027s easy to integrate x to the 10th."},{"Start":"08:26.650 ","End":"08:29.530","Text":"It\u0027s easy to differentiate natural log of x,"},{"Start":"08:29.530 ","End":"08:33.190","Text":"but the integral of natural log of x is not immediate."},{"Start":"08:33.190 ","End":"08:43.735","Text":"The natural choice is to call this one u and to let this bit be dv,"},{"Start":"08:43.735 ","End":"08:46.820","Text":"and that will work fine."},{"Start":"08:50.040 ","End":"08:57.910","Text":"I\u0027d like to use place holders to say that it\u0027s u times v and just look at the formula,"},{"Start":"08:57.910 ","End":"09:04.360","Text":"minus the integral of v and du."},{"Start":"09:04.360 ","End":"09:12.175","Text":"U is natural log of x. V,"},{"Start":"09:12.175 ","End":"09:15.505","Text":"we can get from this by integrating,"},{"Start":"09:15.505 ","End":"09:22.959","Text":"so it\u0027s x to the 11th over 11 according to the formula for powers."},{"Start":"09:22.959 ","End":"09:28.360","Text":"Again, x to the 11 over 11 because again it\u0027s v and du,"},{"Start":"09:28.360 ","End":"09:30.595","Text":"is what we get by differentiating,"},{"Start":"09:30.595 ","End":"09:35.230","Text":"so it\u0027s 1 over x dx."},{"Start":"09:35.230 ","End":"09:37.480","Text":"After a bit of simplification,"},{"Start":"09:37.480 ","End":"09:41.725","Text":"we get that this is equal to 1 11th."},{"Start":"09:41.725 ","End":"09:45.430","Text":"I\u0027ll put the x to the 11th first,"},{"Start":"09:45.430 ","End":"09:51.025","Text":"x to the 11th natural log of x minus."},{"Start":"09:51.025 ","End":"09:52.570","Text":"Now the 1 over 11,"},{"Start":"09:52.570 ","End":"09:55.390","Text":"I\u0027ll take outside the integral sign,"},{"Start":"09:55.390 ","End":"09:59.845","Text":"and it\u0027s the integral of x to the 11th over x,"},{"Start":"09:59.845 ","End":"10:07.010","Text":"which is just x to the 10th dx."},{"Start":"10:08.250 ","End":"10:11.200","Text":"We now have to compute this integral."},{"Start":"10:11.200 ","End":"10:15.475","Text":"This is a lot easier because there\u0027s no extra bit besides the exponent."},{"Start":"10:15.475 ","End":"10:22.210","Text":"We get 1 over 11x to the 11th natural log of x minus."},{"Start":"10:22.210 ","End":"10:26.185","Text":"The integral of this as x to the 11th over 11,"},{"Start":"10:26.185 ","End":"10:30.795","Text":"so it\u0027s minus 1 over 11"},{"Start":"10:30.795 ","End":"10:36.745","Text":"times x to the 11 over 11 plus a constant."},{"Start":"10:36.745 ","End":"10:43.870","Text":"Here I would multiply 11 by 11 and say that this is x to the 11th over a 121,"},{"Start":"10:43.870 ","End":"10:45.640","Text":"but this is okay,"},{"Start":"10:45.640 ","End":"10:47.830","Text":"we can leave it like this."},{"Start":"10:47.830 ","End":"10:53.725","Text":"That\u0027s another example, and it shows that sometimes we have to go with u and v,"},{"Start":"10:53.725 ","End":"10:59.110","Text":"and experience shows which to choose as u and v and there are some rules of them."},{"Start":"10:59.110 ","End":"11:03.400","Text":"Next thing, I\u0027m going to clear the board and then I\u0027ll talk about guidelines or"},{"Start":"11:03.400 ","End":"11:09.890","Text":"rules for how to know which to choose as u and which to choose as dv."},{"Start":"11:10.410 ","End":"11:15.940","Text":"Now before I start discussing which is u and which is dv,"},{"Start":"11:15.940 ","End":"11:18.355","Text":"I want to ask a more basic question."},{"Start":"11:18.355 ","End":"11:21.880","Text":"When should we use integration by parts at all?"},{"Start":"11:21.880 ","End":"11:29.545","Text":"Let me give you a general rule if you have the integral of a polynomial in x."},{"Start":"11:29.545 ","End":"11:35.860","Text":"We have a polynomial multiplied by some calculator function."},{"Start":"11:35.860 ","End":"11:38.605","Text":"I\u0027ll give examples dx."},{"Start":"11:38.605 ","End":"11:43.495","Text":"In this case, I would advise trying integration by parts,"},{"Start":"11:43.495 ","End":"11:46.285","Text":"and it will probably cover,"},{"Start":"11:46.285 ","End":"11:50.005","Text":"say, 75 percent of the cases."},{"Start":"11:50.005 ","End":"11:53.335","Text":"Let me give some examples of what I mean by this."},{"Start":"11:53.335 ","End":"11:59.215","Text":"The integral of x times cosine x dx."},{"Start":"11:59.215 ","End":"12:02.290","Text":"Here I have polynomial just x."},{"Start":"12:02.290 ","End":"12:04.645","Text":"This is a calculator function."},{"Start":"12:04.645 ","End":"12:12.585","Text":"Another example might be the integral of x squared sine x dx,"},{"Start":"12:12.585 ","End":"12:17.055","Text":"again, polynomial times calculator function."},{"Start":"12:17.055 ","End":"12:24.070","Text":"We could have the integral of x squared plus x plus 1 times,"},{"Start":"12:24.070 ","End":"12:28.075","Text":"say, natural log of x dx."},{"Start":"12:28.075 ","End":"12:30.070","Text":"Again, here\u0027s the polynomial,"},{"Start":"12:30.070 ","End":"12:32.665","Text":"and here is the calculator function."},{"Start":"12:32.665 ","End":"12:38.140","Text":"Another example would be the integral of x to"},{"Start":"12:38.140 ","End":"12:46.255","Text":"the 4th times e to the x dx polynomial calculator function."},{"Start":"12:46.255 ","End":"12:55.060","Text":"Then final example, integral of x times arc tangent of x dx."},{"Start":"12:55.060 ","End":"12:58.405","Text":"Once again, polynomial times the calculator function."},{"Start":"12:58.405 ","End":"13:05.860","Text":"There are other classic cases which will use integration by parts."},{"Start":"13:05.860 ","End":"13:08.275","Text":"I\u0027ll give you a couple of examples."},{"Start":"13:08.275 ","End":"13:18.365","Text":"If you see an exponent times a trigonometric like e to the x, cosine x dx,"},{"Start":"13:18.365 ","End":"13:24.010","Text":"or the integral of e to the x times sine x dx,"},{"Start":"13:24.010 ","End":"13:25.510","Text":"or even some variation of this,"},{"Start":"13:25.510 ","End":"13:28.900","Text":"it could be e to the 4x cosine 2x and so on,"},{"Start":"13:28.900 ","End":"13:33.010","Text":"then you should also use integration by parts."},{"Start":"13:33.010 ","End":"13:36.190","Text":"These unusual ones are covered in"},{"Start":"13:36.190 ","End":"13:40.930","Text":"the exercises and you really must do the exercises to fully"},{"Start":"13:40.930 ","End":"13:49.765","Text":"understand the question of which part is u and which part is dv when we have a product."},{"Start":"13:49.765 ","End":"13:52.045","Text":"Let me give some rules for that."},{"Start":"13:52.045 ","End":"13:53.725","Text":"The rule is quite simple."},{"Start":"13:53.725 ","End":"13:56.140","Text":"We look at the calculator function,"},{"Start":"13:56.140 ","End":"13:59.275","Text":"and if it has an immediate integral,"},{"Start":"13:59.275 ","End":"14:02.740","Text":"then this part will be dv."},{"Start":"14:02.740 ","End":"14:05.860","Text":"If not, then it will be u."},{"Start":"14:05.860 ","End":"14:10.510","Text":"For example, we have an immediate integral for cosine x,"},{"Start":"14:10.510 ","End":"14:15.595","Text":"so here this will be dv and this will be u."},{"Start":"14:15.595 ","End":"14:20.260","Text":"Here also, I know how to do the integral of sine x."},{"Start":"14:20.260 ","End":"14:24.160","Text":"This part is going to be dv and this part will be u."},{"Start":"14:24.160 ","End":"14:29.500","Text":"In this one, the calculator function e to the x. I know how to do an immediate integrals."},{"Start":"14:29.500 ","End":"14:32.845","Text":"Again here, this is dv and this is u,"},{"Start":"14:32.845 ","End":"14:37.075","Text":"but in this case and in this case with the natural log and with the arc tangent,"},{"Start":"14:37.075 ","End":"14:40.044","Text":"I don\u0027t know how to integrate these immediately,"},{"Start":"14:40.044 ","End":"14:41.680","Text":"so it\u0027s the other way around."},{"Start":"14:41.680 ","End":"14:45.505","Text":"This becomes u, this becomes u,"},{"Start":"14:45.505 ","End":"14:49.090","Text":"and this with this becomes dv."},{"Start":"14:49.090 ","End":"14:52.840","Text":"Here xdx is dv."},{"Start":"14:52.840 ","End":"14:57.445","Text":"In the case of exponential times in trigonometric,"},{"Start":"14:57.445 ","End":"14:59.470","Text":"it actually doesn\u0027t matter."},{"Start":"14:59.470 ","End":"15:04.525","Text":"Both ways will work and there will be examples of this."},{"Start":"15:04.525 ","End":"15:10.135","Text":"If you answer all the exercises at the end of the chapter following the tutorial,"},{"Start":"15:10.135 ","End":"15:12.775","Text":"if you can solve all of those,"},{"Start":"15:12.775 ","End":"15:15.970","Text":"then you\u0027ll be prepared for pretty much anything they can throw at you,"},{"Start":"15:15.970 ","End":"15:19.359","Text":"at least 99 percent of the cases."},{"Start":"15:19.359 ","End":"15:22.550","Text":"I\u0027m basically done."}],"ID":4412},{"Watched":false,"Name":"Integration by Parts (continued)","Duration":"2m 43s","ChapterTopicVideoID":8325,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.235","Text":"There\u0027s something else I have to say."},{"Start":"00:02.235 ","End":"00:07.935","Text":"There is an alternative notation used in some books and by some teachers."},{"Start":"00:07.935 ","End":"00:09.645","Text":"It\u0027s very similar to this."},{"Start":"00:09.645 ","End":"00:11.219","Text":"It goes as follows."},{"Start":"00:11.219 ","End":"00:17.295","Text":"The integral of uv prime dx,"},{"Start":"00:17.295 ","End":"00:18.885","Text":"u and v are functions of x,"},{"Start":"00:18.885 ","End":"00:28.125","Text":"is equal to uv minus the integral of vu prime dx."},{"Start":"00:28.125 ","End":"00:34.230","Text":"In some sense, this is the same as this because what is du after all but u prime dx,"},{"Start":"00:34.230 ","End":"00:37.305","Text":"and what is dv but v prime dx."},{"Start":"00:37.305 ","End":"00:41.070","Text":"But this way, we don\u0027t deal with the dx,"},{"Start":"00:41.070 ","End":"00:50.250","Text":"we just have a u and v. What we do is take the u and differentiate it and get u prime,"},{"Start":"00:50.250 ","End":"00:55.685","Text":"and the v prime is integrated to get v. It\u0027s just a slightly different notation."},{"Start":"00:55.685 ","End":"01:01.055","Text":"Nevertheless, I\u0027ll do an example with this alternative notation."},{"Start":"01:01.055 ","End":"01:06.680","Text":"The example I\u0027ll take will be the integral of x cubed,"},{"Start":"01:06.680 ","End":"01:10.190","Text":"natural log of x dx."},{"Start":"01:10.190 ","End":"01:13.550","Text":"In this case, we leave aside the dx and just"},{"Start":"01:13.550 ","End":"01:17.500","Text":"decide which part is u and which part is v prime."},{"Start":"01:17.500 ","End":"01:19.340","Text":"The same rules apply."},{"Start":"01:19.340 ","End":"01:23.000","Text":"This is a polynomial times a calculator function."},{"Start":"01:23.000 ","End":"01:26.675","Text":"I don\u0027t know the immediate integral of the natural log."},{"Start":"01:26.675 ","End":"01:28.550","Text":"I want to differentiate this."},{"Start":"01:28.550 ","End":"01:31.535","Text":"I call this 1 u,"},{"Start":"01:31.535 ","End":"01:34.700","Text":"and this 1, I call v prime,"},{"Start":"01:34.700 ","End":"01:38.130","Text":"and then I get that here I want to put u,"},{"Start":"01:38.130 ","End":"01:42.500","Text":"here I want to put v minus the integral."},{"Start":"01:42.500 ","End":"01:45.140","Text":"Here, I want to put v again, and here,"},{"Start":"01:45.140 ","End":"01:49.365","Text":"I want to put u prime, and here dx."},{"Start":"01:49.365 ","End":"01:52.965","Text":"U is natural log of x. V,"},{"Start":"01:52.965 ","End":"01:56.000","Text":"we don\u0027t have, but we have v prime as x cubed,"},{"Start":"01:56.000 ","End":"01:57.500","Text":"so v is the integral of that,"},{"Start":"01:57.500 ","End":"02:00.470","Text":"so it\u0027s x^4 over 4,"},{"Start":"02:00.470 ","End":"02:04.840","Text":"and then again, v, x^4 over 4."},{"Start":"02:04.840 ","End":"02:07.520","Text":"Now, u prime is just the derivative of u,"},{"Start":"02:07.520 ","End":"02:09.574","Text":"which is 1 over x."},{"Start":"02:09.574 ","End":"02:14.205","Text":"We get 1 quarter of"},{"Start":"02:14.205 ","End":"02:20.445","Text":"x^4 natural log of x minus the 1/4 I\u0027ll take outside,"},{"Start":"02:20.445 ","End":"02:24.980","Text":"and I\u0027m left with the integral of x^4 over x,"},{"Start":"02:24.980 ","End":"02:28.555","Text":"which is just x cubed dx."},{"Start":"02:28.555 ","End":"02:34.095","Text":"Then finally, I get here also x^4 over 4,"},{"Start":"02:34.095 ","End":"02:40.410","Text":"so it becomes 1/16th x^4 plus a constant,"},{"Start":"02:40.410 ","End":"02:44.020","Text":"and now, I\u0027m really done."}],"ID":8496},{"Watched":false,"Name":"Exercise 1","Duration":"1m 12s","ChapterTopicVideoID":6699,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.165","Text":"In this exercise we have to compute the following integral,"},{"Start":"00:03.165 ","End":"00:04.710","Text":"and I\u0027m going to do it by parts."},{"Start":"00:04.710 ","End":"00:09.660","Text":"Remember that the formula for integration by parts is that the integral of"},{"Start":"00:09.660 ","End":"00:16.365","Text":"udv equals uv minus the integral of vdu."},{"Start":"00:16.365 ","End":"00:18.300","Text":"Typically, when you have"},{"Start":"00:18.300 ","End":"00:22.395","Text":"a polynomial times a calculator function which you can integrate,"},{"Start":"00:22.395 ","End":"00:24.330","Text":"then that one should be dv."},{"Start":"00:24.330 ","End":"00:30.075","Text":"So this is our dv and this is going to be our u."},{"Start":"00:30.075 ","End":"00:34.260","Text":"So u we have already is x,"},{"Start":"00:34.260 ","End":"00:37.365","Text":"v, we have dv,"},{"Start":"00:37.365 ","End":"00:39.705","Text":"and to get from dv to v it\u0027s the integral."},{"Start":"00:39.705 ","End":"00:44.940","Text":"So it\u0027s just e to the x minus the"},{"Start":"00:44.940 ","End":"00:51.495","Text":"integral of v and we just did v so it\u0027s e to the x,"},{"Start":"00:51.495 ","End":"00:54.900","Text":"du is the derivative,"},{"Start":"00:54.900 ","End":"00:56.550","Text":"if x is u,"},{"Start":"00:56.550 ","End":"00:59.280","Text":"then du is dx."},{"Start":"00:59.280 ","End":"01:03.880","Text":"Now, this stays as is minus,"},{"Start":"01:03.880 ","End":"01:07.685","Text":"the integral of e to the x is just e to the x,"},{"Start":"01:07.685 ","End":"01:13.620","Text":"and in the end we add the constant. We\u0027re done."}],"ID":6760},{"Watched":false,"Name":"Exercise 2","Duration":"2m 1s","ChapterTopicVideoID":6700,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"We have this integral to solve and I\u0027m going to do it by parts."},{"Start":"00:04.890 ","End":"00:09.270","Text":"Remember the formula that the integral of"},{"Start":"00:09.270 ","End":"00:16.695","Text":"udv is equal to uv minus the integral of vdu."},{"Start":"00:16.695 ","End":"00:21.285","Text":"The question is, which here is u and which here is v?"},{"Start":"00:21.285 ","End":"00:24.180","Text":"Natural log of x, we know how to differentiate,"},{"Start":"00:24.180 ","End":"00:30.165","Text":"but we don\u0027t know how to integrate it and that really forces us to take this as our u,"},{"Start":"00:30.165 ","End":"00:37.995","Text":"which means that x to the 4th with the dx is our dv."},{"Start":"00:37.995 ","End":"00:45.285","Text":"This equals u, we have already, times v, well,"},{"Start":"00:45.285 ","End":"00:55.769","Text":"we have dv so v is the integral of this so it\u0027s 1/5x to the 5th minus the integral of v,"},{"Start":"00:55.769 ","End":"01:01.210","Text":"once again, 1/5x to the 5th."},{"Start":"01:01.430 ","End":"01:07.640","Text":"All we need now is du and since u is natural log of x,"},{"Start":"01:07.640 ","End":"01:11.785","Text":"du is 1 over xdx."},{"Start":"01:11.785 ","End":"01:16.120","Text":"We get, let\u0027s put the 5th out front,"},{"Start":"01:16.430 ","End":"01:21.270","Text":"1/5x to the 5th natural log of x minus,"},{"Start":"01:21.270 ","End":"01:24.660","Text":"now I\u0027ll take the 1/5 in front of"},{"Start":"01:24.660 ","End":"01:30.930","Text":"the integral sign and x to the 5th over x is x to the 4th."},{"Start":"01:30.930 ","End":"01:33.350","Text":"I still haven\u0027t done the integral I\u0027ve just simplified,"},{"Start":"01:33.350 ","End":"01:35.225","Text":"but now it\u0027s an easy integral."},{"Start":"01:35.225 ","End":"01:38.060","Text":"This I copy once again 1/5x to"},{"Start":"01:38.060 ","End":"01:45.530","Text":"the 5th natural log of x minus the integral of x to the 4th is x to the 5th over 5."},{"Start":"01:45.530 ","End":"01:53.630","Text":"The over 5 makes it 1 over 25x to the 5th and plus a constant."},{"Start":"01:53.630 ","End":"01:59.000","Text":"If you want, you could take the x to the 5th outside the brackets so 1/5x to the 5th,"},{"Start":"01:59.000 ","End":"02:02.190","Text":"I leave it as that. Done."}],"ID":6761},{"Watched":false,"Name":"Exercise 3","Duration":"1m 21s","ChapterTopicVideoID":6701,"CourseChapterTopicPlaylistID":3684,"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":"We have this integral to solve,"},{"Start":"00:02.100 ","End":"00:05.325","Text":"and it looks like a case for integration by parts."},{"Start":"00:05.325 ","End":"00:07.395","Text":"I\u0027ll remind you of the formula."},{"Start":"00:07.395 ","End":"00:11.310","Text":"We have that the integral of udv"},{"Start":"00:11.310 ","End":"00:18.660","Text":"is equal to uv minus the integral of vdu."},{"Start":"00:18.660 ","End":"00:20.130","Text":"In this case, we have a product,"},{"Start":"00:20.130 ","End":"00:22.230","Text":"polynomial times calculator function"},{"Start":"00:22.230 ","End":"00:23.549","Text":"which we can integrate,"},{"Start":"00:23.549 ","End":"00:28.380","Text":"so we prefer to take this as the dv,"},{"Start":"00:28.380 ","End":"00:31.635","Text":"and I\u0027ll take x as u."},{"Start":"00:31.635 ","End":"00:35.040","Text":"What we get is uv."},{"Start":"00:35.040 ","End":"00:38.370","Text":"U is x, what is v?"},{"Start":"00:38.370 ","End":"00:40.400","Text":"Dv is sine x,"},{"Start":"00:40.400 ","End":"00:43.820","Text":"so v is minus cosine x."},{"Start":"00:43.820 ","End":"00:48.479","Text":"We get x times minus cosine x,"},{"Start":"00:48.479 ","End":"00:52.400","Text":"and then minus the integral v"},{"Start":"00:52.400 ","End":"00:56.090","Text":"is once again minus cosine x."},{"Start":"00:56.090 ","End":"00:57.755","Text":"Then we need du."},{"Start":"00:57.755 ","End":"01:01.670","Text":"If x is u, then dx is du."},{"Start":"01:01.670 ","End":"01:04.040","Text":"That\u0027s all there is to it."},{"Start":"01:04.040 ","End":"01:06.000","Text":"Now, we have to do the integration of course."},{"Start":"01:06.000 ","End":"01:10.220","Text":"This is equal to minus x cosine x."},{"Start":"01:10.220 ","End":"01:13.099","Text":"The minus with the minus is the plus."},{"Start":"01:13.099 ","End":"01:16.580","Text":"Now, the integral of cosine x is sine x,"},{"Start":"01:16.580 ","End":"01:19.515","Text":"and don\u0027t forget to add plus c at the end."},{"Start":"01:19.515 ","End":"01:21.760","Text":"We are done."}],"ID":6762},{"Watched":false,"Name":"Exercise 4","Duration":"2m 28s","ChapterTopicVideoID":6702,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"Here we have this integration to do."},{"Start":"00:02.760 ","End":"00:05.955","Text":"It looks like a case for integration by parts."},{"Start":"00:05.955 ","End":"00:08.040","Text":"Remember the formula that"},{"Start":"00:08.040 ","End":"00:17.175","Text":"the integral of udv is equal to uv minus the integral of vdu."},{"Start":"00:17.175 ","End":"00:19.950","Text":"The main question is, which is u and which is v?"},{"Start":"00:19.950 ","End":"00:22.275","Text":"Polynomial times calculator function."},{"Start":"00:22.275 ","End":"00:25.050","Text":"But we don\u0027t know how to integrate this."},{"Start":"00:25.050 ","End":"00:26.760","Text":"We know how to differentiate this."},{"Start":"00:26.760 ","End":"00:29.144","Text":"So this is going to be u,"},{"Start":"00:29.144 ","End":"00:34.575","Text":"and the rest of it, the polynomial with the dx will be our dv."},{"Start":"00:34.575 ","End":"00:37.500","Text":"So we will get u,"},{"Start":"00:37.500 ","End":"00:41.940","Text":"which is this as is, natural log of x."},{"Start":"00:41.940 ","End":"00:45.065","Text":"V, we get from dv."},{"Start":"00:45.065 ","End":"00:49.805","Text":"If this is dv, then v is the integral polynomial."},{"Start":"00:49.805 ","End":"00:59.835","Text":"It\u0027s 1/3x cubed plus x squared plus 3x minus,"},{"Start":"00:59.835 ","End":"01:01.665","Text":"and now we have the integral."},{"Start":"01:01.665 ","End":"01:04.965","Text":"Again, v, so I just copy that from here,"},{"Start":"01:04.965 ","End":"01:12.800","Text":"so it\u0027s 1/3x cubed plus x squared plus 3x times du."},{"Start":"01:12.800 ","End":"01:14.105","Text":"We don\u0027t have that yet,"},{"Start":"01:14.105 ","End":"01:20.460","Text":"but u is the natural log of x then it\u0027s 1 over x dx."},{"Start":"01:21.700 ","End":"01:24.409","Text":"I have to copy this again."},{"Start":"01:24.409 ","End":"01:31.405","Text":"Natural log of x 1/3x cube plus x squared plus 3x minus,"},{"Start":"01:31.405 ","End":"01:35.975","Text":"and here, I\u0027m going to simplify because we have 1 over x."},{"Start":"01:35.975 ","End":"01:38.870","Text":"So I\u0027m just going to knock down the exponents by 1."},{"Start":"01:38.870 ","End":"01:45.155","Text":"So we get 1/3x squared plus x plus 3,"},{"Start":"01:45.155 ","End":"01:51.810","Text":"the integral of this, dx, and finally the answer."},{"Start":"01:51.810 ","End":"02:02.575","Text":"Again, copy this natural log of x times 1/3x cubed plus x squared plus 3x minus,"},{"Start":"02:02.575 ","End":"02:12.175","Text":"raise to the power by 1 is 3 and divide by it, 1/9x cubed."},{"Start":"02:12.175 ","End":"02:13.440","Text":"I\u0027ll put it in brackets,"},{"Start":"02:13.440 ","End":"02:15.260","Text":"so you don\u0027t have to worry about the minuses."},{"Start":"02:15.260 ","End":"02:20.285","Text":"Integral of x is 1/2x squared or x squared over 2."},{"Start":"02:20.285 ","End":"02:24.260","Text":"Integral of 3 is 3x,"},{"Start":"02:24.260 ","End":"02:28.770","Text":"and let\u0027s not forget the plus C, and we\u0027re done."}],"ID":6763},{"Watched":false,"Name":"Exercise 5","Duration":"1m 33s","ChapterTopicVideoID":6703,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"We have to solve this integral,"},{"Start":"00:02.790 ","End":"00:05.730","Text":"which looks like a case for integration by parts."},{"Start":"00:05.730 ","End":"00:08.580","Text":"I\u0027ll remind you what the formula for that is."},{"Start":"00:08.580 ","End":"00:17.670","Text":"The integral of udv is equal to uv minus the integral of vdu."},{"Start":"00:17.670 ","End":"00:20.595","Text":"The question is, which is u and which is dv?"},{"Start":"00:20.595 ","End":"00:23.785","Text":"Polynomial? Calculate a function,"},{"Start":"00:23.785 ","End":"00:33.515","Text":"we can integrate it that makes this one dv and this one is u. Interpreting this,"},{"Start":"00:33.515 ","End":"00:36.825","Text":"u is equal to x. V,"},{"Start":"00:36.825 ","End":"00:40.410","Text":"we get by integrating cosine 2x."},{"Start":"00:40.410 ","End":"00:44.840","Text":"Since the integral of cosine is sine, if it\u0027s cosine 2x,"},{"Start":"00:44.840 ","End":"00:51.575","Text":"it\u0027s going to be a half sine 2x minus the integral."},{"Start":"00:51.575 ","End":"00:56.085","Text":"Now again, v, which is a half sine 2x,"},{"Start":"00:56.085 ","End":"01:01.305","Text":"and then times du and du is just dx."},{"Start":"01:01.305 ","End":"01:04.550","Text":"This is just a straightforward integral of a sine."},{"Start":"01:04.550 ","End":"01:06.020","Text":"Let\u0027s copy this."},{"Start":"01:06.020 ","End":"01:11.820","Text":"1/2x sine 2x."},{"Start":"01:11.820 ","End":"01:20.600","Text":"Now the integral of sine is minus cosine and we also have to divide again by the half."},{"Start":"01:20.600 ","End":"01:23.450","Text":"The minus with the minus will give me plus and"},{"Start":"01:23.450 ","End":"01:26.195","Text":"the half with the half will give me a quarter."},{"Start":"01:26.195 ","End":"01:28.220","Text":"It\u0027ll be plus a quarter,"},{"Start":"01:28.220 ","End":"01:31.455","Text":"cosine 2x and at the end,"},{"Start":"01:31.455 ","End":"01:34.540","Text":"plus c. We\u0027re done."}],"ID":6764},{"Watched":false,"Name":"Exercise 6","Duration":"4m 4s","ChapterTopicVideoID":6704,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"We have to compute this integral."},{"Start":"00:02.850 ","End":"00:05.550","Text":"Looks like a case for integration by parts."},{"Start":"00:05.550 ","End":"00:07.695","Text":"I\u0027ll remind you of the formula."},{"Start":"00:07.695 ","End":"00:13.590","Text":"The integral of udv is equal to uv,"},{"Start":"00:13.590 ","End":"00:18.015","Text":"minus the integral of vdu."},{"Start":"00:18.015 ","End":"00:24.915","Text":"Now, I will choose dv as sine 4xdx,"},{"Start":"00:24.915 ","End":"00:27.810","Text":"because it\u0027s a calculator function and I\u0027m able to"},{"Start":"00:27.810 ","End":"00:31.065","Text":"integrate it and I prefer to differentiate the polynomial."},{"Start":"00:31.065 ","End":"00:33.750","Text":"This 1 will be u and then,"},{"Start":"00:33.750 ","End":"00:36.930","Text":"we\u0027ll have by the formula, uv."},{"Start":"00:36.930 ","End":"00:38.535","Text":"X squared is u,"},{"Start":"00:38.535 ","End":"00:40.735","Text":"v is the integral of this."},{"Start":"00:40.735 ","End":"00:44.210","Text":"The integral of sine is minus cosine and because of the 4,"},{"Start":"00:44.210 ","End":"00:49.495","Text":"we get a minus a quarter cosine 4x,"},{"Start":"00:49.495 ","End":"00:51.980","Text":"that\u0027s my dv, sorry,"},{"Start":"00:51.980 ","End":"00:53.900","Text":"x squared is times that,"},{"Start":"00:53.900 ","End":"00:55.775","Text":"I\u0027ll just put that in brackets."},{"Start":"00:55.775 ","End":"00:59.615","Text":"Product and then minus the integral."},{"Start":"00:59.615 ","End":"01:04.640","Text":"Once again v, which is minus a quarter"},{"Start":"01:04.640 ","End":"01:10.445","Text":"cosine 4x times du and what is du?"},{"Start":"01:10.445 ","End":"01:13.710","Text":"U is x squared so du is 2xdx."},{"Start":"01:16.580 ","End":"01:19.425","Text":"Let\u0027s see what we get from this."},{"Start":"01:19.425 ","End":"01:24.020","Text":"We get minus 1/4x squared"},{"Start":"01:24.020 ","End":"01:30.975","Text":"cosine 4x and then the 2 with the 1/4 is 1/2,"},{"Start":"01:30.975 ","End":"01:32.655","Text":"minus minus is plus,"},{"Start":"01:32.655 ","End":"01:38.520","Text":"plus 1/2 of the integral of,"},{"Start":"01:38.520 ","End":"01:40.395","Text":"let\u0027s put the x in front,"},{"Start":"01:40.395 ","End":"01:45.580","Text":"so we have x cosine 4xdx."},{"Start":"01:46.790 ","End":"01:50.559","Text":"This is not immediately clear."},{"Start":"01:50.559 ","End":"01:54.650","Text":"Turns out we have to do integration by parts yet again."},{"Start":"01:54.650 ","End":"01:57.320","Text":"I\u0027ll do this as a separate exercise and call"},{"Start":"01:57.320 ","End":"02:03.995","Text":"this exercise asterisk and I\u0027ll do the asterisk at the side here."},{"Start":"02:03.995 ","End":"02:13.165","Text":"We have the integral of x cosine 4xdx."},{"Start":"02:13.165 ","End":"02:16.750","Text":"Once again, we\u0027ll be using this formula, to different u and v,"},{"Start":"02:16.750 ","End":"02:25.055","Text":"it\u0027s the different exercise that will take this bit again as dv and the first bit as u,"},{"Start":"02:25.055 ","End":"02:30.255","Text":"and so we get by the formula uv, u is x,"},{"Start":"02:30.255 ","End":"02:40.620","Text":"v this time will be 1/4 sine 4x minus the integral again, v,"},{"Start":"02:40.620 ","End":"02:44.400","Text":"which is 1/4 sine 4x,"},{"Start":"02:44.400 ","End":"02:46.665","Text":"and then we want du,"},{"Start":"02:46.665 ","End":"02:56.965","Text":"which is just dx and this equals 1/4x sine 4x."},{"Start":"02:56.965 ","End":"03:04.880","Text":"The integral of sine is minus cosine and we\u0027ll also going to get another 1/4."},{"Start":"03:04.880 ","End":"03:07.145","Text":"Together minus, minus and 1/4 and 1/4,"},{"Start":"03:07.145 ","End":"03:12.665","Text":"we\u0027ll get plus 1/16 cosine 4x."},{"Start":"03:12.665 ","End":"03:15.185","Text":"We\u0027ll leave the constant for the very end."},{"Start":"03:15.185 ","End":"03:18.890","Text":"This is the asterisk bit and all I have to do now"},{"Start":"03:18.890 ","End":"03:22.895","Text":"is substitute that in here and add the constant."},{"Start":"03:22.895 ","End":"03:26.645","Text":"From here, I\u0027m going all the way down here."},{"Start":"03:26.645 ","End":"03:35.770","Text":"We get minus 1/4x squared cosine 4x,"},{"Start":"03:35.770 ","End":"03:39.570","Text":"plus 1/2 of the asterisk."},{"Start":"03:39.570 ","End":"03:42.615","Text":"Now, all of this is the asterisk."},{"Start":"03:42.615 ","End":"03:51.340","Text":"1/2 of this, so I get 1/8x sine 4x,"},{"Start":"03:51.680 ","End":"03:57.195","Text":"and then again, 1/2 of the other bits so that makes it"},{"Start":"03:57.195 ","End":"04:04.930","Text":"1/32 cosine of 4x and then plus the constant."}],"ID":6765},{"Watched":false,"Name":"Exercise 7","Duration":"3m 41s","ChapterTopicVideoID":4409,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.795","Text":"We have to compute this integral and I\u0027m going to do it by parts."},{"Start":"00:03.795 ","End":"00:07.815","Text":"I\u0027ll remind you of the formula that the integral of"},{"Start":"00:07.815 ","End":"00:15.000","Text":"u dv is equal to uv minus the integral of v du."},{"Start":"00:15.000 ","End":"00:19.950","Text":"In this case, I\u0027m going to take this bit as the dv,"},{"Start":"00:19.950 ","End":"00:22.770","Text":"and this is going to be du."},{"Start":"00:22.770 ","End":"00:27.675","Text":"What we get is uv x squared."},{"Start":"00:27.675 ","End":"00:29.295","Text":"V we don\u0027t have yet,"},{"Start":"00:29.295 ","End":"00:31.500","Text":"we have to integrate this."},{"Start":"00:31.500 ","End":"00:34.960","Text":"Clearly that\u0027s going to be minus a quarter"},{"Start":"00:34.960 ","End":"00:40.280","Text":"times e to the minus 4x because the internal derivative here is minus 4,"},{"Start":"00:40.280 ","End":"00:42.325","Text":"so we have to divide by it."},{"Start":"00:42.325 ","End":"00:44.805","Text":"That\u0027s uv."},{"Start":"00:44.805 ","End":"00:48.360","Text":"Now, minus the integral of v du."},{"Start":"00:48.360 ","End":"00:55.919","Text":"This is v minus 1/4 e to the minus 4x times du."},{"Start":"00:55.919 ","End":"00:57.345","Text":"U is x squared,"},{"Start":"00:57.345 ","End":"01:02.490","Text":"du is 2x dx."},{"Start":"01:02.490 ","End":"01:09.845","Text":"What we get here is minus a quarter x squared e to the minus 4x."},{"Start":"01:09.845 ","End":"01:12.665","Text":"If I collect the constants in front,"},{"Start":"01:12.665 ","End":"01:23.525","Text":"I get plus 1/2 the integral of xe to the minus 4x dx."},{"Start":"01:23.525 ","End":"01:27.305","Text":"We\u0027re going to have to use integration by parts again."},{"Start":"01:27.305 ","End":"01:29.360","Text":"We still have the formula up."},{"Start":"01:29.360 ","End":"01:34.370","Text":"This time we\u0027re going to let this be dv,"},{"Start":"01:34.370 ","End":"01:35.930","Text":"not the same dv as this."},{"Start":"01:35.930 ","End":"01:37.340","Text":"Should really use different letters,"},{"Start":"01:37.340 ","End":"01:40.985","Text":"but there\u0027s no confusion, and this will be u."},{"Start":"01:40.985 ","End":"01:44.110","Text":"What we\u0027re going to get, and you know what,"},{"Start":"01:44.110 ","End":"01:46.880","Text":"I\u0027ll do this bit as just the integral part,"},{"Start":"01:46.880 ","End":"01:51.485","Text":"this part b asterisk the part of the integral,"},{"Start":"01:51.485 ","End":"01:53.650","Text":"I\u0027ll do it at the side."},{"Start":"01:53.650 ","End":"01:58.140","Text":"The integral of xe to the minus 4x dx,"},{"Start":"01:58.140 ","End":"02:00.600","Text":"I\u0027ve copied just the asterisk."},{"Start":"02:00.600 ","End":"02:05.235","Text":"This we said was u and this part here was dv."},{"Start":"02:05.235 ","End":"02:11.445","Text":"It\u0027s equal to uv, so it\u0027s x. V is 1 minus 1/4,"},{"Start":"02:11.445 ","End":"02:16.435","Text":"sorry, minus 1/4 e to the minus 4x."},{"Start":"02:16.435 ","End":"02:20.870","Text":"That\u0027s v minus the integral of v,"},{"Start":"02:20.870 ","End":"02:25.880","Text":"which is again minus 1/4 e to the minus 4x."},{"Start":"02:25.880 ","End":"02:27.830","Text":"I need du."},{"Start":"02:27.830 ","End":"02:31.625","Text":"This time du is just dx."},{"Start":"02:31.625 ","End":"02:38.230","Text":"This equals minus 1/4 xe to the minus 4x."},{"Start":"02:38.230 ","End":"02:41.360","Text":"Then if we integrate this,"},{"Start":"02:41.360 ","End":"02:44.690","Text":"we\u0027ll get another minus 1/4."},{"Start":"02:44.690 ","End":"02:46.370","Text":"Altogether we\u0027ll get minus, minus,"},{"Start":"02:46.370 ","End":"02:48.515","Text":"minus, and 1/4 and 1/4."},{"Start":"02:48.515 ","End":"02:54.440","Text":"It\u0027ll be minus 1/16 e to the minus 4x."},{"Start":"02:54.440 ","End":"02:57.755","Text":"That\u0027s the asterisk part."},{"Start":"02:57.755 ","End":"03:00.410","Text":"If I put it in here,"},{"Start":"03:00.410 ","End":"03:03.125","Text":"I\u0027m just going to substitute the asterisk,"},{"Start":"03:03.125 ","End":"03:04.580","Text":"which is down here."},{"Start":"03:04.580 ","End":"03:10.160","Text":"We\u0027ll continue this, minus 1/4x squared e to"},{"Start":"03:10.160 ","End":"03:16.430","Text":"the minus 4x plus 1/2 of the asterisk."},{"Start":"03:16.430 ","End":"03:21.620","Text":"So 1/2 times minus 1/4 is minus 1/8,"},{"Start":"03:21.620 ","End":"03:29.180","Text":"so not plus but minus an 1/8 of xe to the minus 4x."},{"Start":"03:29.180 ","End":"03:34.670","Text":"Then again, 1/2 times minus 1/16 is minus"},{"Start":"03:34.670 ","End":"03:42.360","Text":"1 over 32 e to the minus 4x and at the end plus c. We\u0027re done."}],"ID":4420},{"Watched":false,"Name":"Exercise 8","Duration":"1m 53s","ChapterTopicVideoID":4410,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.535","Text":"Here, we have to compute this integral,"},{"Start":"00:02.535 ","End":"00:05.040","Text":"and I\u0027m going to do it by parts."},{"Start":"00:05.040 ","End":"00:08.460","Text":"I\u0027ll write down the formula for integration by parts."},{"Start":"00:08.460 ","End":"00:17.695","Text":"The integral of udv is equal to uv minus the integral of vdu."},{"Start":"00:17.695 ","End":"00:20.965","Text":"Now, the 1 thing you might say is that normally,"},{"Start":"00:20.965 ","End":"00:24.630","Text":"integration by parts involves a product and here,"},{"Start":"00:24.630 ","End":"00:26.294","Text":"we don\u0027t have a product."},{"Start":"00:26.294 ","End":"00:27.810","Text":"Well, in a way, yes,"},{"Start":"00:27.810 ","End":"00:30.660","Text":"we do because I\u0027m going to add a 1 here,"},{"Start":"00:30.660 ","End":"00:35.880","Text":"and I\u0027m going to make it 1 times natural log of x and now, it\u0027s a product."},{"Start":"00:35.880 ","End":"00:40.300","Text":"Now, which is going to be the du and which is going to be dv?"},{"Start":"00:40.300 ","End":"00:43.310","Text":"I don\u0027t know the integral of natural log of x,"},{"Start":"00:43.310 ","End":"00:45.140","Text":"but I do know how to differentiate it."},{"Start":"00:45.140 ","End":"00:50.690","Text":"That makes this 1 a natural choice to be the u and the 1dx,"},{"Start":"00:50.690 ","End":"00:55.300","Text":"or just the dx, will be the dv,"},{"Start":"00:55.300 ","End":"00:58.860","Text":"so we get, by the formula, u,"},{"Start":"00:58.860 ","End":"01:03.990","Text":"which is this, times v. We don\u0027t have that yet,"},{"Start":"01:03.990 ","End":"01:07.910","Text":"but if dv is dx, then v is x,"},{"Start":"01:07.910 ","End":"01:13.130","Text":"so natural log of x times x minus the integral of v,"},{"Start":"01:13.130 ","End":"01:16.295","Text":"which is x and du."},{"Start":"01:16.295 ","End":"01:18.260","Text":"Well, if u is natural log of x,"},{"Start":"01:18.260 ","End":"01:20.465","Text":"then du is 1 over x dx,"},{"Start":"01:20.465 ","End":"01:23.810","Text":"so 1 over x dx."},{"Start":"01:23.810 ","End":"01:25.775","Text":"This brings us to,"},{"Start":"01:25.775 ","End":"01:29.195","Text":"I prefer to write dx in front, less confusing,"},{"Start":"01:29.195 ","End":"01:33.799","Text":"x natural log of x minus the integral,"},{"Start":"01:33.799 ","End":"01:38.340","Text":"x times 1 over x is just 1, is 1 dx."},{"Start":"01:38.340 ","End":"01:40.540","Text":"I don\u0027t even have to write the 1."},{"Start":"01:40.540 ","End":"01:44.240","Text":"In any event, x natural log of x,"},{"Start":"01:44.240 ","End":"01:46.580","Text":"the integral of 1 is just x,"},{"Start":"01:46.580 ","End":"01:49.594","Text":"so it\u0027s minus x plus a constant,"},{"Start":"01:49.594 ","End":"01:53.850","Text":"and that\u0027s the integral of natural log of x. Done."}],"ID":4421},{"Watched":false,"Name":"Exercise 9","Duration":"1m 54s","ChapterTopicVideoID":4411,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"In this exercise, we have to compute"},{"Start":"00:02.070 ","End":"00:04.950","Text":"the integral of the natural log of 1"},{"Start":"00:04.950 ","End":"00:07.410","Text":"over the cube root of x dx."},{"Start":"00:07.410 ","End":"00:10.050","Text":"I suggest we do a bit of algebra first"},{"Start":"00:10.050 ","End":"00:12.900","Text":"and remember our rules of exponents."},{"Start":"00:12.900 ","End":"00:16.290","Text":"1 over the cube root of x"},{"Start":"00:16.290 ","End":"00:20.670","Text":"is in fact 1 over x to the power of 1/3,"},{"Start":"00:20.670 ","End":"00:23.325","Text":"because the cube root means to the power of 1/3."},{"Start":"00:23.325 ","End":"00:25.155","Text":"Now, 1 over something,"},{"Start":"00:25.155 ","End":"00:27.630","Text":"we can rewrite that with a negative exponent"},{"Start":"00:27.630 ","End":"00:30.705","Text":"as x to the power of minus 1/3."},{"Start":"00:30.705 ","End":"00:32.595","Text":"If I do this here,"},{"Start":"00:32.595 ","End":"00:34.680","Text":"I\u0027ll get that this is equal to the integral"},{"Start":"00:34.680 ","End":"00:42.055","Text":"of the natural log of x to the power of minus 1/3 dx."},{"Start":"00:42.055 ","End":"00:44.750","Text":"Now, another thing I\u0027d like you to recall"},{"Start":"00:44.750 ","End":"00:48.905","Text":"is the rules of natural logarithm or any logarithm."},{"Start":"00:48.905 ","End":"00:50.870","Text":"The logarithm of an exponent,"},{"Start":"00:50.870 ","End":"00:52.730","Text":"for example, the logarithm"},{"Start":"00:52.730 ","End":"00:56.360","Text":"of general a to the power of n"},{"Start":"00:56.360 ","End":"01:00.530","Text":"is n times the logarithm of a to b."},{"Start":"01:00.530 ","End":"01:02.420","Text":"Natural logarithm could be any base."},{"Start":"01:02.420 ","End":"01:04.550","Text":"If I apply this here,"},{"Start":"01:04.550 ","End":"01:09.260","Text":"then we\u0027ll get that the logarithm of x"},{"Start":"01:09.260 ","End":"01:15.260","Text":"to the minus 1/3 is equal to minus 1/3 logarithm of x,"},{"Start":"01:15.260 ","End":"01:17.165","Text":"natural log in this case."},{"Start":"01:17.165 ","End":"01:19.100","Text":"At this point, we get"},{"Start":"01:19.100 ","End":"01:22.470","Text":"and taking the minus 1/3 in front"},{"Start":"01:22.470 ","End":"01:24.300","Text":"of the integral all in 1 step"},{"Start":"01:24.300 ","End":"01:29.250","Text":"is minus 1/3 the integral of natural log of x dx."},{"Start":"01:30.020 ","End":"01:32.360","Text":"Now, normally I would say,"},{"Start":"01:32.360 ","End":"01:34.985","Text":"we should do this with integration by parts."},{"Start":"01:34.985 ","End":"01:36.470","Text":"But because it\u0027s already appeared"},{"Start":"01:36.470 ","End":"01:38.150","Text":"in a previous exercise"},{"Start":"01:38.150 ","End":"01:39.785","Text":"and we have the solution,"},{"Start":"01:39.785 ","End":"01:42.140","Text":"I\u0027m just going to write down"},{"Start":"01:42.140 ","End":"01:43.845","Text":"what the solution was,"},{"Start":"01:43.845 ","End":"01:48.920","Text":"and it happened to be x natural log of x minus x,"},{"Start":"01:48.920 ","End":"01:50.675","Text":"if I recall correctly."},{"Start":"01:50.675 ","End":"01:52.100","Text":"At this point, we have to add"},{"Start":"01:52.100 ","End":"01:55.560","Text":"the constant of integration and we\u0027re done."}],"ID":4422},{"Watched":false,"Name":"Exercise 10","Duration":"2m 51s","ChapterTopicVideoID":4412,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"Here, we have to compute the integral of the arc tangent and I\u0027m going to do it by parts."},{"Start":"00:05.280 ","End":"00:07.575","Text":"First of all, I\u0027ll remind you of the formula."},{"Start":"00:07.575 ","End":"00:15.975","Text":"The integral of udv is equal to uv minus the integral of vdu."},{"Start":"00:15.975 ","End":"00:19.230","Text":"The question is, what\u0027s u and what\u0027s dv?"},{"Start":"00:19.230 ","End":"00:20.985","Text":"Usually, it involves a product,"},{"Start":"00:20.985 ","End":"00:22.260","Text":"but if we don\u0027t see a product,"},{"Start":"00:22.260 ","End":"00:24.790","Text":"the product could just be 1 times the function."},{"Start":"00:24.790 ","End":"00:27.585","Text":"Now, we don\u0027t want to integrate arc tangent."},{"Start":"00:27.585 ","End":"00:29.160","Text":"That\u0027s the whole exercise."},{"Start":"00:29.160 ","End":"00:30.770","Text":"We\u0027re going to want to differentiate this,"},{"Start":"00:30.770 ","End":"00:34.490","Text":"which makes this the natural choice to be our u,"},{"Start":"00:34.490 ","End":"00:43.235","Text":"and then dx will be dv or 1dx if you like to have another function, but like this,"},{"Start":"00:43.235 ","End":"00:45.484","Text":"and then by the formula,"},{"Start":"00:45.484 ","End":"00:50.315","Text":"this is going to equal u is arc tangent of"},{"Start":"00:50.315 ","End":"00:58.785","Text":"x and dv is dx so v is equal to just x."},{"Start":"00:58.785 ","End":"01:07.670","Text":"This is arc tangent x times x minus the integral of v. Again,"},{"Start":"01:07.670 ","End":"01:15.650","Text":"v is x and the u is just the derivative of this dx."},{"Start":"01:15.650 ","End":"01:25.205","Text":"The derivative of arc tangent is 1 over 1 plus x squared dx,"},{"Start":"01:25.205 ","End":"01:32.010","Text":"and I\u0027ll just rewrite that as x times the arc tangent of x,"},{"Start":"01:32.010 ","End":"01:36.455","Text":"and I won\u0027t need the brackets, minus the integral."},{"Start":"01:36.455 ","End":"01:38.450","Text":"I\u0027ll put the x on top of here,"},{"Start":"01:38.450 ","End":"01:44.575","Text":"so it\u0027s x over 1 plus x squared dx."},{"Start":"01:44.575 ","End":"01:52.955","Text":"Now, this almost looks like one of those formulas that the integral of f prime over"},{"Start":"01:52.955 ","End":"01:58.535","Text":"f dx is equal to the natural log of"},{"Start":"01:58.535 ","End":"02:04.670","Text":"f but the derivative of the denominator is not quite the numerator."},{"Start":"02:04.670 ","End":"02:06.275","Text":"If I had 2x here,"},{"Start":"02:06.275 ","End":"02:07.625","Text":"that would be better."},{"Start":"02:07.625 ","End":"02:10.250","Text":"So why not? Let\u0027s just write a 2 here,"},{"Start":"02:10.250 ","End":"02:11.600","Text":"but hey, that\u0027s cheating."},{"Start":"02:11.600 ","End":"02:14.060","Text":"You can\u0027t just add a 2, so if I put 1/2 here,"},{"Start":"02:14.060 ","End":"02:16.145","Text":"I think everything will be okay."},{"Start":"02:16.145 ","End":"02:21.155","Text":"Now, we can say that this is equal to x"},{"Start":"02:21.155 ","End":"02:27.835","Text":"times the arc tangent of x minus 1/2,"},{"Start":"02:27.835 ","End":"02:31.265","Text":"and using this formula with f being the denominator,"},{"Start":"02:31.265 ","End":"02:35.690","Text":"I get the natural log of 1 plus x squared."},{"Start":"02:35.690 ","End":"02:37.850","Text":"Well, actually it should be absolute value here,"},{"Start":"02:37.850 ","End":"02:40.280","Text":"so I\u0027ll put 1 plus x squared,"},{"Start":"02:40.280 ","End":"02:44.200","Text":"although that was unnecessary because this is in any event positive."},{"Start":"02:44.200 ","End":"02:46.620","Text":"I\u0027ll write it like this, and of course,"},{"Start":"02:46.620 ","End":"02:47.820","Text":"we add the plus,"},{"Start":"02:47.820 ","End":"02:49.395","Text":"the constant, at the end,"},{"Start":"02:49.395 ","End":"02:51.760","Text":"and that\u0027s the answer."}],"ID":4423},{"Watched":false,"Name":"Exercise 11","Duration":"4m 52s","ChapterTopicVideoID":4413,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.575","Text":"In this exercise, we have to compute the integral of arc sine of x."},{"Start":"00:04.575 ","End":"00:08.310","Text":"It\u0027s similar to a previous one we did with the arc tangent."},{"Start":"00:08.310 ","End":"00:10.980","Text":"In that we\u0027re going to use integration by parts,"},{"Start":"00:10.980 ","End":"00:18.795","Text":"and the formula for that is the integral of u dv equals uv minus the integral of vdu,"},{"Start":"00:18.795 ","End":"00:23.160","Text":"and this part is going to be du because we know how to differentiate this,"},{"Start":"00:23.160 ","End":"00:25.740","Text":"and this will be dv."},{"Start":"00:25.740 ","End":"00:30.545","Text":"What I\u0027m going to get, and I\u0027ll write it on the next line, is u,"},{"Start":"00:30.545 ","End":"00:35.704","Text":"which is arc sine of x,"},{"Start":"00:35.704 ","End":"00:39.860","Text":"and then v is going to be x,"},{"Start":"00:39.860 ","End":"00:42.815","Text":"because if dv is dx, then v is just x,"},{"Start":"00:42.815 ","End":"00:48.860","Text":"so times x and then minus the integral v,"},{"Start":"00:48.860 ","End":"00:54.215","Text":"as we said, is x and du is just the derivative of this, dx."},{"Start":"00:54.215 ","End":"00:58.660","Text":"Now the derivative of the arc sine is 1"},{"Start":"00:58.660 ","End":"01:07.135","Text":"over the square root of 1 minus x squared dx."},{"Start":"01:07.135 ","End":"01:11.000","Text":"Now, how are we going to do this integral?"},{"Start":"01:11.000 ","End":"01:15.815","Text":"Well, experience shows that this will work by substitution,"},{"Start":"01:15.815 ","End":"01:22.085","Text":"if I substitute t is equal to 1 minus x squared."},{"Start":"01:22.085 ","End":"01:23.755","Text":"What makes me confident?"},{"Start":"01:23.755 ","End":"01:26.615","Text":"Because I already almost have the t."},{"Start":"01:26.615 ","End":"01:30.905","Text":"The t would be minus 2x dx."},{"Start":"01:30.905 ","End":"01:34.010","Text":"Now I don\u0027t have minus 2x, I have minus x,"},{"Start":"01:34.010 ","End":"01:36.170","Text":"but a factor of 2 or a constant,"},{"Start":"01:36.170 ","End":"01:37.700","Text":"and is not going to make much difference."},{"Start":"01:37.700 ","End":"01:40.070","Text":"So this looks like it should work."},{"Start":"01:40.070 ","End":"01:43.774","Text":"If t is equal to 1 minus x squared,"},{"Start":"01:43.774 ","End":"01:49.365","Text":"then dt is equal to minus 2x dx,"},{"Start":"01:49.365 ","End":"01:59.640","Text":"and I recommend writing dx is equal to dt over minus 2x."},{"Start":"01:59.640 ","End":"02:02.490","Text":"Then we can substitute dx."},{"Start":"02:02.490 ","End":"02:06.180","Text":"We can also substitute t here,"},{"Start":"02:06.180 ","End":"02:08.145","Text":"and then let\u0027s see what we get."},{"Start":"02:08.145 ","End":"02:12.950","Text":"This is going to equal arc sine of x."},{"Start":"02:12.950 ","End":"02:16.985","Text":"Actually, why don\u0027t I put the x in front,"},{"Start":"02:16.985 ","End":"02:18.890","Text":"then I won\u0027t need brackets."},{"Start":"02:18.890 ","End":"02:21.770","Text":"This should really go with brackets because it\u0027s going to be confusing."},{"Start":"02:21.770 ","End":"02:26.645","Text":"So it\u0027s x arc sine of x minus. Now let\u0027s see."},{"Start":"02:26.645 ","End":"02:35.045","Text":"We have the integral of x times 1 over the square root of t,"},{"Start":"02:35.045 ","End":"02:43.680","Text":"because t is 1 minus x squared and dx is dt over minus 2x."},{"Start":"02:43.680 ","End":"02:49.280","Text":"This x will cancel with this x and well,"},{"Start":"02:49.280 ","End":"02:51.040","Text":"the minus will cancel with the minus,"},{"Start":"02:51.040 ","End":"02:52.670","Text":"so that will be a plus."},{"Start":"02:52.670 ","End":"02:54.020","Text":"What I\u0027ll be left with,"},{"Start":"02:54.020 ","End":"02:56.960","Text":"if I take 2 outside the brackets,"},{"Start":"02:56.960 ","End":"03:03.180","Text":"is x arc sine x. I\u0027m just dragging this along."},{"Start":"03:03.180 ","End":"03:06.180","Text":"Plus the 2 comes out,"},{"Start":"03:06.180 ","End":"03:07.710","Text":"so it\u0027s one-half,"},{"Start":"03:07.710 ","End":"03:16.650","Text":"the integral of dt over the square root of t. Now,"},{"Start":"03:16.650 ","End":"03:17.920","Text":"I just noticed something,"},{"Start":"03:17.920 ","End":"03:21.710","Text":"that we have a square root of t in the denominator and I happen to remember,"},{"Start":"03:21.710 ","End":"03:22.880","Text":"and you should too,"},{"Start":"03:22.880 ","End":"03:25.145","Text":"to meet one of those immediate derivatives,"},{"Start":"03:25.145 ","End":"03:31.090","Text":"that the derivative of the square root is 1 over twice the square root."},{"Start":"03:31.090 ","End":"03:33.540","Text":"Really, I should have left the 2 here,"},{"Start":"03:33.540 ","End":"03:35.580","Text":"and I\u0027m going to just do that."},{"Start":"03:35.580 ","End":"03:38.010","Text":"Here\u0027s the 2 inside."},{"Start":"03:38.010 ","End":"03:40.685","Text":"Then I can say that this is equal to."},{"Start":"03:40.685 ","End":"03:44.785","Text":"I don\u0027t know why I\u0027m dragging this all the way with me, but that\u0027s okay."},{"Start":"03:44.785 ","End":"03:46.405","Text":"Not really too much bother."},{"Start":"03:46.405 ","End":"03:49.120","Text":"Arc sine x plus,"},{"Start":"03:49.120 ","End":"03:54.790","Text":"and the integral of this will be now the square root of t plus a constant."},{"Start":"03:54.790 ","End":"03:58.915","Text":"But I have to go from t back to x,"},{"Start":"03:58.915 ","End":"04:01.030","Text":"and here I use this."},{"Start":"04:01.030 ","End":"04:08.080","Text":"I finally get that this is equal to x times arc sine of"},{"Start":"04:08.080 ","End":"04:16.970","Text":"x plus the square root of 1 minus x squared plus a constant."},{"Start":"04:17.180 ","End":"04:22.745","Text":"That\u0027s it. Like to point out it could have been done with other techniques."},{"Start":"04:22.745 ","End":"04:26.000","Text":"At this point, we could have seen that the derivative of 1"},{"Start":"04:26.000 ","End":"04:29.795","Text":"minus x squared is almost what we have here,"},{"Start":"04:29.795 ","End":"04:31.504","Text":"x except for a constant,"},{"Start":"04:31.504 ","End":"04:32.810","Text":"and then we could have used the method,"},{"Start":"04:32.810 ","End":"04:34.040","Text":"I forget what they call it,"},{"Start":"04:34.040 ","End":"04:37.235","Text":"to derivative already inside or something,"},{"Start":"04:37.235 ","End":"04:38.900","Text":"that could have been done that way."},{"Start":"04:38.900 ","End":"04:43.460","Text":"It also could have been done by substitution with a different substitution."},{"Start":"04:43.460 ","End":"04:46.360","Text":"For example, t equals square root of 1 minus x,"},{"Start":"04:46.360 ","End":"04:48.500","Text":"but you might like to try that as an exercise and"},{"Start":"04:48.500 ","End":"04:50.900","Text":"that will also give you the same answer."},{"Start":"04:50.900 ","End":"04:53.100","Text":"Anyway, I\u0027m done."}],"ID":4424},{"Watched":false,"Name":"Exercise 12","Duration":"8m 13s","ChapterTopicVideoID":4414,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Here we have to compute the following integral."},{"Start":"00:02.700 ","End":"00:06.720","Text":"Notice that it contains a fifth root and a natural logarithm."},{"Start":"00:06.720 ","End":"00:08.330","Text":"We\u0027ve seen something similar before,"},{"Start":"00:08.330 ","End":"00:10.335","Text":"it\u0027s time to do a bit of algebra."},{"Start":"00:10.335 ","End":"00:16.200","Text":"For one thing, the fifth root of something, in this case,"},{"Start":"00:16.200 ","End":"00:19.290","Text":"x minus 2 is simply that thing to the power of a fifth,"},{"Start":"00:19.290 ","End":"00:22.290","Text":"so we have x minus 2 to the power of fifth."},{"Start":"00:22.290 ","End":"00:24.990","Text":"Then we have a natural logarithm."},{"Start":"00:24.990 ","End":"00:27.180","Text":"The natural logarithm of an exponent,"},{"Start":"00:27.180 ","End":"00:29.280","Text":"and we\u0027ve already done this kind of thing before,"},{"Start":"00:29.280 ","End":"00:33.540","Text":"of x minus 2 to the power of 1/5."},{"Start":"00:33.540 ","End":"00:41.505","Text":"We just take the 1/5 in front of the logarithm and it\u0027s 1/5 the logarithm of x minus 2."},{"Start":"00:41.505 ","End":"00:45.200","Text":"If we put this back in here,"},{"Start":"00:45.200 ","End":"00:48.965","Text":"what we\u0027re going to get is that this is equal to,"},{"Start":"00:48.965 ","End":"00:53.375","Text":"now the fifth can come in front of the integral sign, so it\u0027s 1/5,"},{"Start":"00:53.375 ","End":"01:04.055","Text":"the integral of x times the natural logarithm of x minus 2 dx."},{"Start":"01:04.055 ","End":"01:09.185","Text":"Now, this we\u0027re going to do with integration by parts."},{"Start":"01:09.185 ","End":"01:14.555","Text":"Let me remind you of the formula of integration by parts,"},{"Start":"01:14.555 ","End":"01:18.085","Text":"and that is the integral of"},{"Start":"01:18.085 ","End":"01:26.225","Text":"udv is equal to uv minus the integral of vdu."},{"Start":"01:26.225 ","End":"01:29.570","Text":"I don\u0027t want to keep dragging the 1/5 with me throughout."},{"Start":"01:29.570 ","End":"01:32.960","Text":"Let\u0027s just take the integral of this bit"},{"Start":"01:32.960 ","End":"01:37.310","Text":"first and at the end we\u0027ll remember to do the 1/5."},{"Start":"01:37.310 ","End":"01:47.340","Text":"I\u0027m continuing down here with the integral of x times natural log of x minus 2 dx."},{"Start":"01:47.340 ","End":"01:51.729","Text":"It\u0027s pretty clear that we want to take this as u,"},{"Start":"01:51.729 ","End":"01:56.745","Text":"and this will be our dv."},{"Start":"01:56.745 ","End":"02:05.730","Text":"First we need the uv, now u is natural log of x minus 2, but what is v?"},{"Start":"02:05.730 ","End":"02:07.245","Text":"We don\u0027t quite have v,"},{"Start":"02:07.245 ","End":"02:11.580","Text":"we know that dv is xdx."},{"Start":"02:11.580 ","End":"02:15.590","Text":"In that case, by taking the integral or primitive,"},{"Start":"02:15.590 ","End":"02:19.610","Text":"we get that v is 1/2x squared."},{"Start":"02:19.610 ","End":"02:22.295","Text":"We don\u0027t need the constant at this point."},{"Start":"02:22.295 ","End":"02:26.720","Text":"Times 1/2 of x squared, now,"},{"Start":"02:26.720 ","End":"02:31.550","Text":"that\u0027s uv minus the integral of v,"},{"Start":"02:31.550 ","End":"02:34.780","Text":"which is again 1/2x squared."},{"Start":"02:34.780 ","End":"02:39.065","Text":"Du is going to be the derivative."},{"Start":"02:39.065 ","End":"02:42.575","Text":"Du will be the derivative of this,"},{"Start":"02:42.575 ","End":"02:50.370","Text":"which is 1 over x minus 2 times the inner derivative which is just 1."},{"Start":"02:50.370 ","End":"02:57.765","Text":"Here it\u0027s a 1/2x squared times du times 1 over x minus 2,"},{"Start":"02:57.765 ","End":"03:01.575","Text":"and dx, of course, dx here."},{"Start":"03:01.575 ","End":"03:05.010","Text":"Just going to slightly rewrite, simplify,"},{"Start":"03:05.010 ","End":"03:09.665","Text":"so this is equal to 1/2x squared"},{"Start":"03:09.665 ","End":"03:15.120","Text":"times natural log of x minus 2 minus,"},{"Start":"03:15.120 ","End":"03:18.480","Text":"and I\u0027ll take the 1/2 out in front of the integral,"},{"Start":"03:18.480 ","End":"03:20.639","Text":"put the x squared on top of the fraction,"},{"Start":"03:20.639 ","End":"03:24.885","Text":"x squared over x minus 2 dx."},{"Start":"03:24.885 ","End":"03:29.105","Text":"Now what we\u0027re left with is this integral,"},{"Start":"03:29.105 ","End":"03:31.415","Text":"x squared over x minus 2."},{"Start":"03:31.415 ","End":"03:34.550","Text":"Now, there are many ways of doing this,"},{"Start":"03:34.550 ","End":"03:38.030","Text":"one way would be long division of polynomials,"},{"Start":"03:38.030 ","End":"03:41.225","Text":"because a degree here is higher than the degree here."},{"Start":"03:41.225 ","End":"03:43.610","Text":"The other way would be a substitution."},{"Start":"03:43.610 ","End":"03:46.580","Text":"We could substitute t equals x minus 2."},{"Start":"03:46.580 ","End":"03:48.575","Text":"But I\u0027m going to choose a third way,"},{"Start":"03:48.575 ","End":"03:51.020","Text":"which is using a little trick."},{"Start":"03:51.020 ","End":"03:55.530","Text":"What I\u0027m going to do, well first I\u0027ll copy this beginning bit,"},{"Start":"03:55.530 ","End":"03:58.415","Text":"which has nothing to do with the integral."},{"Start":"03:58.415 ","End":"04:01.085","Text":"But if you look at what I\u0027m doing,"},{"Start":"04:01.085 ","End":"04:03.785","Text":"I\u0027m going to say instead of x squared,"},{"Start":"04:03.785 ","End":"04:06.770","Text":"I\u0027ll write x squared minus 4."},{"Start":"04:06.770 ","End":"04:10.325","Text":"The denominator will still be x minus 2."},{"Start":"04:10.325 ","End":"04:14.075","Text":"Now, why would x squared minus 4 be any better for me?"},{"Start":"04:14.075 ","End":"04:19.115","Text":"Because x minus 2 goes into this by the difference of square\u0027s law."},{"Start":"04:19.115 ","End":"04:21.335","Text":"But of course I can\u0027t just subtract 4,"},{"Start":"04:21.335 ","End":"04:24.175","Text":"so I also have to add 4."},{"Start":"04:24.175 ","End":"04:34.005","Text":"At this stage, maybe I should just work and I\u0027ll use a different color just on this part."},{"Start":"04:34.005 ","End":"04:36.930","Text":"Again, not to keep dragging everything."},{"Start":"04:36.930 ","End":"04:43.185","Text":"I forgot the dx, of course, dx."},{"Start":"04:43.185 ","End":"04:44.490","Text":"I work on this part,"},{"Start":"04:44.490 ","End":"04:46.335","Text":"then I\u0027ll substitute that in here,"},{"Start":"04:46.335 ","End":"04:47.700","Text":"and when I\u0027ve got all of this,"},{"Start":"04:47.700 ","End":"04:49.890","Text":"then I\u0027ll substitute that in here with the fifth,"},{"Start":"04:49.890 ","End":"04:52.140","Text":"and I think we can sort it out."},{"Start":"04:52.140 ","End":"04:55.175","Text":"I\u0027m just working on the pink part now."},{"Start":"04:55.175 ","End":"05:03.150","Text":"This part is the pink and this part from here is the cyan."},{"Start":"05:03.190 ","End":"05:12.620","Text":"I\u0027m going to break it up into 2, we\u0027re going to have x squared minus 4 over x minus 2 dx,"},{"Start":"05:12.620 ","End":"05:20.460","Text":"plus the integral of 4 over x minus 2 dx."},{"Start":"05:20.460 ","End":"05:22.890","Text":"This is going to be easy now."},{"Start":"05:22.890 ","End":"05:25.085","Text":"This part, if we divide it,"},{"Start":"05:25.085 ","End":"05:32.300","Text":"this x squared minus 4 is equal to x minus 2x plus 2."},{"Start":"05:32.300 ","End":"05:41.595","Text":"This bit is just going to be the integral of x plus 2 dx,"},{"Start":"05:41.595 ","End":"05:50.270","Text":"and this bit is just going to be 4 times the integral of 1 over x minus 2 dx."},{"Start":"05:50.270 ","End":"05:52.480","Text":"This one is immediate,"},{"Start":"05:52.480 ","End":"05:58.410","Text":"x plus 2 gives me 1/2 x squared plus 2x,"},{"Start":"05:58.410 ","End":"06:03.610","Text":"plus, now the integral of 1 over x is natural log of x."},{"Start":"06:03.620 ","End":"06:06.200","Text":"If it\u0027s just x minus 2,"},{"Start":"06:06.200 ","End":"06:08.330","Text":"it\u0027s still internal derivative 1,"},{"Start":"06:08.330 ","End":"06:11.135","Text":"so it\u0027s still natural log of x minus 2,"},{"Start":"06:11.135 ","End":"06:12.260","Text":"but there\u0027s a 4 here,"},{"Start":"06:12.260 ","End":"06:16.760","Text":"so it\u0027s 4 times natural log of x minus 2."},{"Start":"06:16.760 ","End":"06:18.480","Text":"I won\u0027t bother with the absolute value,"},{"Start":"06:18.480 ","End":"06:21.965","Text":"let\u0027s assume that we\u0027re always working with x bigger than 2, say."},{"Start":"06:21.965 ","End":"06:24.020","Text":"Then plus the constant,"},{"Start":"06:24.020 ","End":"06:26.060","Text":"and that is it."},{"Start":"06:26.060 ","End":"06:31.680","Text":"But that\u0027s not the answer because we just done the purple, the magenta part."},{"Start":"06:31.680 ","End":"06:36.230","Text":"Let me just remove that plus C because we really just need to put it once at the end."},{"Start":"06:36.230 ","End":"06:37.820","Text":"That\u0027s the purple part."},{"Start":"06:37.820 ","End":"06:41.975","Text":"Now I need to piece back the cyan turquoise part."},{"Start":"06:41.975 ","End":"06:52.200","Text":"The cyan bit will now be 1/2 x squared natural log of x minus 2,"},{"Start":"06:52.200 ","End":"06:57.360","Text":"minus 1/2 of this bit,"},{"Start":"06:57.360 ","End":"07:07.570","Text":"which is 1/2 x squared plus 2x plus 4 natural log of x minus 2."},{"Start":"07:07.610 ","End":"07:11.130","Text":"That\u0027s just the magenta part,"},{"Start":"07:11.130 ","End":"07:12.500","Text":"but at the end,"},{"Start":"07:12.500 ","End":"07:15.320","Text":"we\u0027re also going to have to get back to the original,"},{"Start":"07:15.320 ","End":"07:17.990","Text":"which is just adding an extra 1/5."},{"Start":"07:17.990 ","End":"07:20.730","Text":"Let\u0027s just put the 1/5."},{"Start":"07:20.730 ","End":"07:25.230","Text":"1/5 times 1/2 is 1/10."},{"Start":"07:25.330 ","End":"07:31.760","Text":"1/10 x squared natural log of x minus 2."},{"Start":"07:31.760 ","End":"07:35.600","Text":"Now, the 1/5 with this 1/2 also becomes 1/10."},{"Start":"07:35.600 ","End":"07:43.385","Text":"1 /10 here, minus 1/20 of x squared,"},{"Start":"07:43.385 ","End":"07:45.515","Text":"because this is like 1/10,"},{"Start":"07:45.515 ","End":"07:47.945","Text":"instead of the 2 you\u0027d think of it as a 10."},{"Start":"07:47.945 ","End":"07:56.980","Text":"Now, 1/10 times 2x is 1/5x,"},{"Start":"07:56.980 ","End":"08:06.515","Text":"and 1/10 of 4 is 4/10 or 2/5 of natural log of x minus 2."},{"Start":"08:06.515 ","End":"08:08.630","Text":"Perhaps it could be tidied up a bit,"},{"Start":"08:08.630 ","End":"08:11.000","Text":"but I\u0027ll just leave it like this."},{"Start":"08:11.000 ","End":"08:13.799","Text":"But we do have to add a constant."}],"ID":4425},{"Watched":false,"Name":"Exercise 13","Duration":"3m 12s","ChapterTopicVideoID":4415,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.849","Text":"In this exercise, we\u0027re asked to find the integral of x over cosine squared x dx."},{"Start":"00:05.849 ","End":"00:08.085","Text":"Now, I would like to do this by parts,"},{"Start":"00:08.085 ","End":"00:12.255","Text":"but integration by parts works better with products and not quotients."},{"Start":"00:12.255 ","End":"00:23.520","Text":"Let\u0027s write this as the integral of x times 1 over cosine squared x dx."},{"Start":"00:23.520 ","End":"00:27.585","Text":"I\u0027d like to remind you of the formula for integration by parts,"},{"Start":"00:27.585 ","End":"00:36.375","Text":"which says that the integral of udv is uv minus the integral of vdu."},{"Start":"00:36.375 ","End":"00:38.010","Text":"The usual question is,"},{"Start":"00:38.010 ","End":"00:40.380","Text":"which is u and which is v?"},{"Start":"00:40.380 ","End":"00:44.675","Text":"The answer is that this is going to be u and this is going to be"},{"Start":"00:44.675 ","End":"00:49.565","Text":"dv because 1 over cosine squared is one of those immediate integrals,"},{"Start":"00:49.565 ","End":"00:53.600","Text":"its integral is the tangent of x. I\u0027m going to call"},{"Start":"00:53.600 ","End":"00:58.860","Text":"this part u and this part is going to be dv."},{"Start":"00:58.860 ","End":"01:01.595","Text":"Then according to the formula,"},{"Start":"01:01.595 ","End":"01:04.670","Text":"I\u0027m going to get uv, so u is x,"},{"Start":"01:04.670 ","End":"01:09.420","Text":"and now if dv is 1 over cosine squared,"},{"Start":"01:09.420 ","End":"01:12.185","Text":"then v is equal to tangent of x."},{"Start":"01:12.185 ","End":"01:15.605","Text":"Derivative of tangent x is 1 over cosine squared."},{"Start":"01:15.605 ","End":"01:19.670","Text":"We get x, which is u times v,"},{"Start":"01:19.670 ","End":"01:26.284","Text":"which is tangent x minus the integral of v,"},{"Start":"01:26.284 ","End":"01:28.910","Text":"which is again tangent x,"},{"Start":"01:28.910 ","End":"01:32.060","Text":"and du, if u is x,"},{"Start":"01:32.060 ","End":"01:34.540","Text":"then du is just dx."},{"Start":"01:34.540 ","End":"01:40.350","Text":"This is equal to x tangent x minus,"},{"Start":"01:40.350 ","End":"01:43.345","Text":"now what I\u0027d like to do is a side exercise,"},{"Start":"01:43.345 ","End":"01:50.135","Text":"I would like to compute this part first below and then come back here."},{"Start":"01:50.135 ","End":"01:57.235","Text":"What I want to do is do the integral of tangent xdx,"},{"Start":"01:57.235 ","End":"01:59.625","Text":"and at the end I\u0027ll come back here."},{"Start":"01:59.625 ","End":"02:02.630","Text":"This is equal to the integral,"},{"Start":"02:02.630 ","End":"02:05.430","Text":"tangent is sine over cosine."},{"Start":"02:05.430 ","End":"02:11.620","Text":"Now, we have this rule that the integral of f prime over"},{"Start":"02:11.620 ","End":"02:18.705","Text":"f is just the natural log of f plus a constant."},{"Start":"02:18.705 ","End":"02:22.030","Text":"In this case, I want to take my f as cosine x,"},{"Start":"02:22.030 ","End":"02:26.755","Text":"but I don\u0027t quite have f prime above because the derivative of cosine is minus sine."},{"Start":"02:26.755 ","End":"02:31.030","Text":"But how about why don\u0027t I just put a minus here and a minus here?"},{"Start":"02:31.030 ","End":"02:32.690","Text":"I think that\u0027ll be okay right?"},{"Start":"02:32.690 ","End":"02:38.650","Text":"Now, I do have the case of cosine x being f minus sine x is f prime,"},{"Start":"02:38.650 ","End":"02:46.440","Text":"so I get minus natural log of cosine of x."},{"Start":"02:46.440 ","End":"02:48.270","Text":"Now that I have this,"},{"Start":"02:48.270 ","End":"02:50.360","Text":"I can come back here,"},{"Start":"02:50.360 ","End":"02:57.050","Text":"so I\u0027ll just go back up there and get that this is x tangent x minus,"},{"Start":"02:57.050 ","End":"03:02.525","Text":"but this minus here with this minus here makes it plus"},{"Start":"03:02.525 ","End":"03:10.010","Text":"natural log of cosine of x and finally plus constant."},{"Start":"03:10.010 ","End":"03:13.080","Text":"This here is the answer."}],"ID":4426},{"Watched":false,"Name":"Exercise 14","Duration":"3m 13s","ChapterTopicVideoID":4416,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.815","Text":"In this exercise, we have to compute the integral natural log of x over x squared,"},{"Start":"00:04.815 ","End":"00:06.690","Text":"I\u0027ve copied it here."},{"Start":"00:06.690 ","End":"00:08.745","Text":"I want to use integration by parts,"},{"Start":"00:08.745 ","End":"00:11.115","Text":"but for that I need a product, not a quotient."},{"Start":"00:11.115 ","End":"00:14.700","Text":"I\u0027ll rewrite this as the integral natural log of"},{"Start":"00:14.700 ","End":"00:18.675","Text":"x and 1 over x squared is just x to the minus 2."},{"Start":"00:18.675 ","End":"00:23.940","Text":"Remember the formula for integration by parts is that the integral of"},{"Start":"00:23.940 ","End":"00:31.665","Text":"udv is equal to uv minus the integral of vdu."},{"Start":"00:31.665 ","End":"00:36.470","Text":"We\u0027re going to take u as a natural log of x because it\u0027s easy to differentiate,"},{"Start":"00:36.470 ","End":"00:42.445","Text":"not easy to integrate or take this is u and the x minus 2 to the minus 2,"},{"Start":"00:42.445 ","End":"00:45.290","Text":"dx will be my dv."},{"Start":"00:45.290 ","End":"00:47.449","Text":"If I apply this formula,"},{"Start":"00:47.449 ","End":"00:51.305","Text":"I will get that this is equal to now uv,"},{"Start":"00:51.305 ","End":"00:54.439","Text":"u is natural log of x,"},{"Start":"00:54.439 ","End":"00:56.675","Text":"but what is v?"},{"Start":"00:56.675 ","End":"01:01.174","Text":"Well, if dv is x to the minus 2 dx,"},{"Start":"01:01.174 ","End":"01:03.350","Text":"then v is the integral of that."},{"Start":"01:03.350 ","End":"01:07.400","Text":"It\u0027s x to the minus 1 over minus 1."},{"Start":"01:07.400 ","End":"01:13.340","Text":"I can just rewrite that as minus x to the minus 1."},{"Start":"01:13.340 ","End":"01:16.005","Text":"I don\u0027t want this minus 1 in the denominator."},{"Start":"01:16.005 ","End":"01:20.890","Text":"We have natural log of x times v,"},{"Start":"01:20.890 ","End":"01:22.770","Text":"which is minus x to the minus 1."},{"Start":"01:22.770 ","End":"01:27.399","Text":"I will put the minus here and then x to the minus 1,"},{"Start":"01:27.399 ","End":"01:32.850","Text":"and then minus the integral of vdu."},{"Start":"01:32.850 ","End":"01:38.840","Text":"Again, v is minus x to the minus 1,"},{"Start":"01:38.840 ","End":"01:43.990","Text":"and du will be 1 over x dx."},{"Start":"01:43.990 ","End":"01:49.905","Text":"If u is natural log of x du is 1 over x dx."},{"Start":"01:49.905 ","End":"01:52.620","Text":"Let\u0027s just simplify this a bit,"},{"Start":"01:52.620 ","End":"01:56.555","Text":"this would be minus natural log of x"},{"Start":"01:56.555 ","End":"02:01.775","Text":"over x. I want to go back to the fractional form and not the negative exponent."},{"Start":"02:01.775 ","End":"02:05.935","Text":"Let\u0027s see now. Minus with a minus will make it plus."},{"Start":"02:05.935 ","End":"02:09.540","Text":"This is 1 over x and this is 1 over x,"},{"Start":"02:09.540 ","End":"02:12.980","Text":"It\u0027s the integral of 1 over x squared dx."},{"Start":"02:12.980 ","End":"02:14.150","Text":"But in this case,"},{"Start":"02:14.150 ","End":"02:16.600","Text":"I actually prefer the x to the minus 2 form."},{"Start":"02:16.600 ","End":"02:19.640","Text":"Let\u0027s just say this 1 over x is also x to the minus 1."},{"Start":"02:19.640 ","End":"02:23.120","Text":"It\u0027s x to the minus 2 dx,"},{"Start":"02:23.120 ","End":"02:28.189","Text":"which equals minus natural log of x over x."},{"Start":"02:28.189 ","End":"02:36.020","Text":"The integral of x to the minus 2 would be x to the minus 1 over minus 1,"},{"Start":"02:36.020 ","End":"02:37.760","Text":"just like we had before,"},{"Start":"02:37.760 ","End":"02:42.185","Text":"which is minus x to the minus 1 as before."},{"Start":"02:42.185 ","End":"02:48.480","Text":"I would like to write it as minus 1 over x finally."},{"Start":"02:48.480 ","End":"02:56.880","Text":"What I have here is minus 1 over x plus a constant,"},{"Start":"02:56.880 ","End":"02:59.180","Text":"and that\u0027ll do as the answer."},{"Start":"02:59.180 ","End":"03:01.850","Text":"If you want to put a common denominator,"},{"Start":"03:01.850 ","End":"03:09.935","Text":"we could say that this is equal to minus natural log of x plus 1 over x plus c,"},{"Start":"03:09.935 ","End":"03:13.770","Text":"not necessary, this would be the answer."}],"ID":4427},{"Watched":false,"Name":"Exercise 15","Duration":"3m 58s","ChapterTopicVideoID":4417,"CourseChapterTopicPlaylistID":3684,"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":"Here we have to compute the integral of x arctangent of xdx."},{"Start":"00:04.020 ","End":"00:06.420","Text":"We\u0027re going to use integration by parts."},{"Start":"00:06.420 ","End":"00:16.090","Text":"Remember the formula that the integral of udv is equal to uv minus the integral of vdu."},{"Start":"00:16.090 ","End":"00:19.680","Text":"Now I\u0027m going to want to differentiate the arctangent."},{"Start":"00:19.680 ","End":"00:22.454","Text":"I\u0027ll let this 1 be u,"},{"Start":"00:22.454 ","End":"00:26.730","Text":"which leaves me that xdx is dv."},{"Start":"00:26.730 ","End":"00:32.220","Text":"Now what I want here is I want to have uv,"},{"Start":"00:32.220 ","End":"00:38.640","Text":"something here u, something here v, minus the integral."},{"Start":"00:38.640 ","End":"00:41.554","Text":"Here I\u0027m going to put v,"},{"Start":"00:41.554 ","End":"00:45.635","Text":"and here I\u0027m going to put du. Let\u0027s see."},{"Start":"00:45.635 ","End":"00:51.830","Text":"U, we already have is arctangent of x. V is going"},{"Start":"00:51.830 ","End":"00:58.575","Text":"to be 1/2 of x squared because dv is xdx and I need to integrate that."},{"Start":"00:58.575 ","End":"01:02.625","Text":"V is going to be 1/2 x squared."},{"Start":"01:02.625 ","End":"01:08.334","Text":"Likewise, here v is 1/2 x squared and du,"},{"Start":"01:08.334 ","End":"01:12.585","Text":"I\u0027ll get by differentiating the arctangent of x."},{"Start":"01:12.585 ","End":"01:15.185","Text":"That\u0027s an immediate derivative,"},{"Start":"01:15.185 ","End":"01:19.800","Text":"and that\u0027s 1 over 1 plus x squared dx."},{"Start":"01:19.800 ","End":"01:23.210","Text":"Simplifying this a little bit we get,"},{"Start":"01:23.210 ","End":"01:25.265","Text":"I\u0027ll put this 1 first,"},{"Start":"01:25.265 ","End":"01:30.765","Text":"1/2 x squared arctangent of x"},{"Start":"01:30.765 ","End":"01:37.575","Text":"minus the 1/2 in front and x squared over the denominator."},{"Start":"01:37.575 ","End":"01:41.115","Text":"This is what we get."},{"Start":"01:41.115 ","End":"01:44.605","Text":"This is the integral that we have to do."},{"Start":"01:44.605 ","End":"01:48.100","Text":"I\u0027ve highlighted the integral part and I want to do this at"},{"Start":"01:48.100 ","End":"01:51.415","Text":"the side and then we\u0027ll come back here and substitute it."},{"Start":"01:51.415 ","End":"01:54.565","Text":"What I want to do now is just the highlighted bit,"},{"Start":"01:54.565 ","End":"02:02.690","Text":"which is the integral of x squared over 1 plus x squared dx."},{"Start":"02:02.690 ","End":"02:07.015","Text":"There\u0027s a standard trick we use in cases like this."},{"Start":"02:07.015 ","End":"02:08.995","Text":"We said that this is the integral."},{"Start":"02:08.995 ","End":"02:11.725","Text":"Now if I had 1 plus x squared here,"},{"Start":"02:11.725 ","End":"02:12.820","Text":"that would be nice,"},{"Start":"02:12.820 ","End":"02:15.204","Text":"because then I would be able to cancel."},{"Start":"02:15.204 ","End":"02:18.820","Text":"But obviously, I can\u0027t just add a 1 plus here."},{"Start":"02:18.820 ","End":"02:23.075","Text":"I also have to put minus 1 to compensate."},{"Start":"02:23.075 ","End":"02:27.670","Text":"Now this minus will allow me to split this integral into 2."},{"Start":"02:27.670 ","End":"02:36.730","Text":"What I\u0027ll get is the integral of 1 plus x squared over 1 plus x"},{"Start":"02:36.730 ","End":"02:46.855","Text":"squared dx minus the integral of 1 over 1 plus x squared dx."},{"Start":"02:46.855 ","End":"02:50.049","Text":"Now, this just cancels."},{"Start":"02:50.049 ","End":"02:53.725","Text":"I mean, this whole thing is just 1."},{"Start":"02:53.725 ","End":"03:00.860","Text":"What I get is the integral of 1 is just x."},{"Start":"03:00.860 ","End":"03:05.260","Text":"The integral of 1 over 1 plus x squared is an immediate integral."},{"Start":"03:05.260 ","End":"03:06.700","Text":"It\u0027s the arctangent. I mean,"},{"Start":"03:06.700 ","End":"03:09.970","Text":"we just used it here that the derivative of this is that."},{"Start":"03:09.970 ","End":"03:19.240","Text":"This will be minus arctangent x. I don\u0027t need to add the plus constant here."},{"Start":"03:19.240 ","End":"03:20.845","Text":"We\u0027ll do that at the end."},{"Start":"03:20.845 ","End":"03:23.455","Text":"This is the highlighted bit."},{"Start":"03:23.455 ","End":"03:31.710","Text":"Now I\u0027m going to go back here to this line and continue over here."},{"Start":"03:31.710 ","End":"03:37.105","Text":"What we get is 1/2, I\u0027m copying from here,"},{"Start":"03:37.105 ","End":"03:46.410","Text":"x squared arctangent of x minus 1/2 of this thing here,"},{"Start":"03:46.410 ","End":"03:51.625","Text":"x minus arctangent x,"},{"Start":"03:51.625 ","End":"03:55.325","Text":"and then plus c. We could simplify this."},{"Start":"03:55.325 ","End":"03:56.780","Text":"I\u0027m not going to bother."},{"Start":"03:56.780 ","End":"03:59.190","Text":"This is it. We\u0027re done."}],"ID":4428},{"Watched":false,"Name":"Exercise 16","Duration":"6m 22s","ChapterTopicVideoID":4418,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this exercise, we have to compute the integral of x squared,"},{"Start":"00:04.110 ","End":"00:07.215","Text":"natural log of x squared plus 1 dx."},{"Start":"00:07.215 ","End":"00:09.300","Text":"We\u0027re going to do it by parts."},{"Start":"00:09.300 ","End":"00:13.470","Text":"I\u0027ll remind you of the formula that the integral of"},{"Start":"00:13.470 ","End":"00:21.595","Text":"udv is equal to uv minus the integral of vdu."},{"Start":"00:21.595 ","End":"00:25.709","Text":"We\u0027re going to want to differentiate the natural log."},{"Start":"00:25.709 ","End":"00:28.620","Text":"This 1 is going to be u,"},{"Start":"00:28.620 ","End":"00:33.615","Text":"and what remains the x squared dx will be our dv."},{"Start":"00:33.615 ","End":"00:36.630","Text":"Now what we want to write here is u,"},{"Start":"00:36.630 ","End":"00:37.830","Text":"and I\u0027ll fill these things in,"},{"Start":"00:37.830 ","End":"00:42.965","Text":"in a moment, times v minus the integral."},{"Start":"00:42.965 ","End":"00:45.665","Text":"Here we\u0027re going to put v,"},{"Start":"00:45.665 ","End":"00:48.140","Text":"and here we\u0027re going to put du."},{"Start":"00:48.140 ","End":"00:50.405","Text":"I\u0027m just copying the formula."},{"Start":"00:50.405 ","End":"00:52.430","Text":"Now, u, we have already,"},{"Start":"00:52.430 ","End":"00:57.935","Text":"which is the natural log of x squared plus 1."},{"Start":"00:57.935 ","End":"01:03.064","Text":"We don\u0027t have v, but if dv is x squared dx,"},{"Start":"01:03.064 ","End":"01:05.030","Text":"then v is the integral of that."},{"Start":"01:05.030 ","End":"01:08.330","Text":"It\u0027s 1 third of x cubed."},{"Start":"01:08.330 ","End":"01:12.795","Text":"Here again, I put v 1 third x cubed."},{"Start":"01:12.795 ","End":"01:14.875","Text":"What I need is du."},{"Start":"01:14.875 ","End":"01:17.705","Text":"Now I have u, which is a natural log."},{"Start":"01:17.705 ","End":"01:23.240","Text":"So du is just the derivative of this over this itself."},{"Start":"01:23.240 ","End":"01:33.570","Text":"What we have is 2x over x squared plus 1 or if you like to think of it,"},{"Start":"01:33.570 ","End":"01:36.710","Text":"it\u0027s 1 over x squared plus 1 times the inner derivative,"},{"Start":"01:36.710 ","End":"01:40.130","Text":"which is 2x and dx."},{"Start":"01:40.130 ","End":"01:44.150","Text":"Just simplify this a bit. Let\u0027s see."},{"Start":"01:44.150 ","End":"01:49.130","Text":"I prefer to reverse the order here and write it as 1 third x cubed,"},{"Start":"01:49.130 ","End":"01:52.490","Text":"natural log of x squared plus 1."},{"Start":"01:52.490 ","End":"01:55.030","Text":"That\u0027s this bit minus."},{"Start":"01:55.030 ","End":"02:00.900","Text":"Here, I\u0027ll take the 2 over 3 outside the integration sign."},{"Start":"02:00.900 ","End":"02:02.815","Text":"I\u0027ve got 2 thirds."},{"Start":"02:02.815 ","End":"02:11.515","Text":"The integral x cubed times x is x to the fourth over x squared plus 1dx."},{"Start":"02:11.515 ","End":"02:14.345","Text":"Just to make things a bit easier,"},{"Start":"02:14.345 ","End":"02:18.380","Text":"we\u0027ll compute just the integral I\u0027ve highlighted here,"},{"Start":"02:18.380 ","End":"02:20.870","Text":"the side, and then we\u0027ll come back here."},{"Start":"02:20.870 ","End":"02:30.000","Text":"We have the integral of x to the fourth over x squared plus 1dx."},{"Start":"02:30.790 ","End":"02:37.025","Text":"I\u0027m going to use 1 of the standard algebraic tricks here to simplify this."},{"Start":"02:37.025 ","End":"02:41.525","Text":"I\u0027m going to write it as the integral of x to the fourth."},{"Start":"02:41.525 ","End":"02:43.820","Text":"Now I\u0027m going to put a minus 1 here."},{"Start":"02:43.820 ","End":"02:45.710","Text":"You\u0027ll see in a moment why,"},{"Start":"02:45.710 ","End":"02:51.830","Text":"and a plus 1 over the same x squared plus 1."},{"Start":"02:51.830 ","End":"02:56.285","Text":"The reason I\u0027m doing this is because this thing is a difference of squares,"},{"Start":"02:56.285 ","End":"03:01.515","Text":"and we\u0027ll be able to use a formula to simplify it. Let\u0027s see."},{"Start":"03:01.515 ","End":"03:03.790","Text":"What we get is,"},{"Start":"03:03.790 ","End":"03:08.765","Text":"remember that a squared minus b squared is a minus b, a plus b."},{"Start":"03:08.765 ","End":"03:13.115","Text":"This is just equal to the integral."},{"Start":"03:13.115 ","End":"03:15.095","Text":"Oh, and I forgot to say."},{"Start":"03:15.095 ","End":"03:18.620","Text":"What I\u0027m going to do is to separate the"},{"Start":"03:18.620 ","End":"03:22.625","Text":"integral into the sum of 2 integrals using this here."},{"Start":"03:22.625 ","End":"03:25.385","Text":"What we get before I break this up,"},{"Start":"03:25.385 ","End":"03:28.760","Text":"and I\u0027ll just write it as x^4th,"},{"Start":"03:28.760 ","End":"03:37.130","Text":"minus 1 over x squared plus 1 dx plus the integral."},{"Start":"03:37.130 ","End":"03:38.825","Text":"This plus is this plus,"},{"Start":"03:38.825 ","End":"03:45.555","Text":"plus 1 over x squared plus 1 dx."},{"Start":"03:45.555 ","End":"03:48.305","Text":"Now, I\u0027m going to use the formula."},{"Start":"03:48.305 ","End":"03:50.120","Text":"Oh, pardon me."},{"Start":"03:50.120 ","End":"03:51.980","Text":"This is x^4th."},{"Start":"03:51.980 ","End":"03:56.690","Text":"This is the bit which I am going to split using the difference of squares"},{"Start":"03:56.690 ","End":"04:01.340","Text":"rule as x squared minus 1 x"},{"Start":"04:01.340 ","End":"04:06.570","Text":"squared plus 1 over x"},{"Start":"04:06.570 ","End":"04:12.090","Text":"squared plus 1 dx plus,"},{"Start":"04:12.090 ","End":"04:14.420","Text":"and just to emphasize it,"},{"Start":"04:14.420 ","End":"04:17.830","Text":"this bit here is this bit here."},{"Start":"04:17.830 ","End":"04:20.840","Text":"I highlighted it to emphasize."},{"Start":"04:20.840 ","End":"04:29.270","Text":"Plus integral of 1 over x squared plus 1 dx."},{"Start":"04:29.270 ","End":"04:36.950","Text":"Now, this x squared plus 1 will cancel with this x squared plus 1."},{"Start":"04:36.950 ","End":"04:44.210","Text":"All will be left with over here is the integral of x"},{"Start":"04:44.210 ","End":"04:52.605","Text":"squared minus 1dx plus the integral of 1 over."},{"Start":"04:52.605 ","End":"04:58.655","Text":"Let me write this as 1 over 1 plus x squared because it looks more familiar."},{"Start":"04:58.655 ","End":"05:00.815","Text":"Why does it look more familiar?"},{"Start":"05:00.815 ","End":"05:02.620","Text":"This is the arctangent,"},{"Start":"05:02.620 ","End":"05:04.085","Text":"but let\u0027s do it in order."},{"Start":"05:04.085 ","End":"05:11.970","Text":"This is equal to integral of x squared is x cubed over 3 minus 1,"},{"Start":"05:11.970 ","End":"05:14.010","Text":"I just get minus x."},{"Start":"05:14.010 ","End":"05:15.500","Text":"Here plus, as I said,"},{"Start":"05:15.500 ","End":"05:16.580","Text":"it\u0027s an immediate integral."},{"Start":"05:16.580 ","End":"05:19.250","Text":"It\u0027s the arctangent of x,"},{"Start":"05:19.250 ","End":"05:24.155","Text":"and finally plus c. This is not the end of the exercise."},{"Start":"05:24.155 ","End":"05:27.290","Text":"Remember, we just did the highlighted."},{"Start":"05:27.290 ","End":"05:35.310","Text":"We have to go back and substitute this instead of the highlighted bit here."},{"Start":"05:35.310 ","End":"05:37.640","Text":"What we\u0027ll get is,"},{"Start":"05:37.640 ","End":"05:39.920","Text":"I\u0027m just reading off this line."},{"Start":"05:39.920 ","End":"05:42.830","Text":"But instead of the highlighted bit I\u0027ll put this."},{"Start":"05:42.830 ","End":"05:50.420","Text":"What we get ultimately is this bit 1/3 x cubed,"},{"Start":"05:50.420 ","End":"05:57.805","Text":"natural log of x squared plus 1 minus 2/3."},{"Start":"05:57.805 ","End":"06:01.920","Text":"All I have to do is just copy this bit actually was redundant."},{"Start":"06:01.920 ","End":"06:04.460","Text":"I\u0027ll need to write the plus c here."},{"Start":"06:04.460 ","End":"06:07.475","Text":"For a moment there I thought we were at the end."},{"Start":"06:07.475 ","End":"06:16.425","Text":"X cubed over 3 minus x plus arctangent x."},{"Start":"06:16.425 ","End":"06:18.740","Text":"Now we really are done."},{"Start":"06:18.740 ","End":"06:23.490","Text":"Now we put the plus constant of integration. That\u0027s it."}],"ID":4429},{"Watched":false,"Name":"Exercise 17","Duration":"2m 40s","ChapterTopicVideoID":4419,"CourseChapterTopicPlaylistID":3684,"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 the square of the natural log of x dx,"},{"Start":"00:05.535 ","End":"00:07.410","Text":"and I\u0027m going to do it by parts."},{"Start":"00:07.410 ","End":"00:09.675","Text":"I\u0027d like to remind you of the formula,"},{"Start":"00:09.675 ","End":"00:14.670","Text":"that the integral of u dv is equal"},{"Start":"00:14.670 ","End":"00:20.820","Text":"to uv minus the integral of v du."},{"Start":"00:20.820 ","End":"00:22.770","Text":"I\u0027m going to want to differentiate this bit,"},{"Start":"00:22.770 ","End":"00:26.265","Text":"so this bit is what I\u0027ll call u,"},{"Start":"00:26.265 ","End":"00:29.340","Text":"and this will be my dv."},{"Start":"00:29.340 ","End":"00:30.960","Text":"What this is going to equal to,"},{"Start":"00:30.960 ","End":"00:32.985","Text":"I\u0027ll just write the general form from here."},{"Start":"00:32.985 ","End":"00:34.675","Text":"I\u0027ll need a u here,"},{"Start":"00:34.675 ","End":"00:37.220","Text":"I\u0027ll need a v here."},{"Start":"00:37.220 ","End":"00:39.995","Text":"I\u0027ll need minus the integral,"},{"Start":"00:39.995 ","End":"00:43.135","Text":"and I\u0027ll need here V again,"},{"Start":"00:43.135 ","End":"00:45.910","Text":"and here I\u0027ll need du."},{"Start":"00:45.910 ","End":"00:49.220","Text":"Let\u0027s see, u we already have."},{"Start":"00:49.220 ","End":"00:52.005","Text":"Natural log of x,"},{"Start":"00:52.005 ","End":"00:54.485","Text":"all squared, we write the 2 just here."},{"Start":"00:54.485 ","End":"00:56.465","Text":"If dv is dx,"},{"Start":"00:56.465 ","End":"01:00.625","Text":"then v is just x times,"},{"Start":"01:00.625 ","End":"01:03.825","Text":"v is x, again,"},{"Start":"01:03.825 ","End":"01:08.855","Text":"and du is going to be the derivative of this dx."},{"Start":"01:08.855 ","End":"01:17.045","Text":"It\u0027s twice natural log of x times the inner derivative,"},{"Start":"01:17.045 ","End":"01:21.275","Text":"which is 1 over x dx."},{"Start":"01:21.275 ","End":"01:23.360","Text":"Simplify this a bit."},{"Start":"01:23.360 ","End":"01:26.210","Text":"What we get is this times this,"},{"Start":"01:26.210 ","End":"01:27.830","Text":"I\u0027ll just write it in reverse order,"},{"Start":"01:27.830 ","End":"01:29.630","Text":"I think it looks better."},{"Start":"01:29.630 ","End":"01:35.150","Text":"X natural log squared of x minus."},{"Start":"01:35.150 ","End":"01:37.865","Text":"Now here, some stuff cancels."},{"Start":"01:37.865 ","End":"01:41.285","Text":"This x cancels with this x."},{"Start":"01:41.285 ","End":"01:45.465","Text":"The 2 I\u0027m going to pull out in front here,"},{"Start":"01:45.465 ","End":"01:51.515","Text":"and actually all we\u0027re left with here is the natural log of x dx."},{"Start":"01:51.515 ","End":"01:55.220","Text":"Now, normally I would just do this at the side,"},{"Start":"01:55.220 ","End":"01:59.210","Text":"but we did this already in a previous exercise,"},{"Start":"01:59.210 ","End":"02:02.450","Text":"and I\u0027ll just remind you what we got there,"},{"Start":"02:02.450 ","End":"02:04.130","Text":"so just this bit,"},{"Start":"02:04.130 ","End":"02:06.635","Text":"like I said from a previous exercise,"},{"Start":"02:06.635 ","End":"02:08.585","Text":"we also did it by parts there,"},{"Start":"02:08.585 ","End":"02:14.570","Text":"came out to be x natural log of x minus x plus a constant."},{"Start":"02:14.570 ","End":"02:16.700","Text":"I\u0027m not going to do this again."},{"Start":"02:16.700 ","End":"02:19.160","Text":"If you\u0027d like, check that the derivative of this is this,"},{"Start":"02:19.160 ","End":"02:24.200","Text":"or do this by parts where this is u and this is dv."},{"Start":"02:24.200 ","End":"02:27.530","Text":"I\u0027m just going to use the result and say that what we"},{"Start":"02:27.530 ","End":"02:31.720","Text":"get here is x natural log squared of x"},{"Start":"02:31.720 ","End":"02:41.190","Text":"minus twice x natural log of x minus x plus a constant, and we\u0027re done."}],"ID":4430},{"Watched":false,"Name":"Exercise 18","Duration":"6m 18s","ChapterTopicVideoID":4420,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"Here we have to compute this integral,"},{"Start":"00:02.610 ","End":"00:05.550","Text":"I\u0027ve copied it here, which I want to do by parts."},{"Start":"00:05.550 ","End":"00:08.010","Text":"But integration by parts likes products,"},{"Start":"00:08.010 ","End":"00:11.430","Text":"not quotients so let me do some algebra and"},{"Start":"00:11.430 ","End":"00:15.615","Text":"write this as the numerator squared over the denominator squared."},{"Start":"00:15.615 ","End":"00:22.690","Text":"It\u0027s natural log squared of x times 1 over x squared."},{"Start":"00:24.230 ","End":"00:30.150","Text":"I\u0027ll just rewrite it once again to use negative exponents."},{"Start":"00:30.150 ","End":"00:41.020","Text":"I have the integral of natural log squared of x times x to the minus 2dx."},{"Start":"00:41.020 ","End":"00:46.730","Text":"The reason I wrote it as x to the minus 2 is that if I integrate or differentiate,"},{"Start":"00:46.730 ","End":"00:48.920","Text":"I have the power formula."},{"Start":"00:48.920 ","End":"00:56.585","Text":"I mentioned integration by parts so let me remind you of the formula that the integral of"},{"Start":"00:56.585 ","End":"01:06.510","Text":"udv is equal to uv minus the integral of vdu."},{"Start":"01:06.510 ","End":"01:08.070","Text":"Now to differentiate this,"},{"Start":"01:08.070 ","End":"01:14.125","Text":"this is going to be my u and this bit will be dv."},{"Start":"01:14.125 ","End":"01:17.690","Text":"This will equal, let me just write the template first from here."},{"Start":"01:17.690 ","End":"01:20.345","Text":"I\u0027m going to need something here which will be u,"},{"Start":"01:20.345 ","End":"01:28.055","Text":"something here which will be v minus the integral of something here which will be v,"},{"Start":"01:28.055 ","End":"01:30.455","Text":"and something here which will be du."},{"Start":"01:30.455 ","End":"01:32.765","Text":"Now let\u0027s see, u we already have,"},{"Start":"01:32.765 ","End":"01:36.800","Text":"natural log squared of x. What about v?"},{"Start":"01:36.800 ","End":"01:38.600","Text":"If dv is this,"},{"Start":"01:38.600 ","End":"01:40.880","Text":"then v is the integral of this,"},{"Start":"01:40.880 ","End":"01:44.690","Text":"which using exponents, we raise the exponent by 1,"},{"Start":"01:44.690 ","End":"01:45.770","Text":"it gives us minus 1,"},{"Start":"01:45.770 ","End":"01:46.970","Text":"and divide by it."},{"Start":"01:46.970 ","End":"01:51.365","Text":"We just get minus x to the minus 1."},{"Start":"01:51.365 ","End":"01:55.320","Text":"I mean it was over minus 1 but put the minus here."},{"Start":"01:55.320 ","End":"01:59.085","Text":"Again, minus x to the minus 1,"},{"Start":"01:59.085 ","End":"02:01.070","Text":"and du I don\u0027t have,"},{"Start":"02:01.070 ","End":"02:03.560","Text":"I just differentiate this using the chain rule."},{"Start":"02:03.560 ","End":"02:08.930","Text":"I get twice natural log of x times inner derivative,"},{"Start":"02:08.930 ","End":"02:10.550","Text":"which is 1 over x,"},{"Start":"02:10.550 ","End":"02:12.800","Text":"and let\u0027s not forget the dx."},{"Start":"02:12.800 ","End":"02:19.085","Text":"A bit of simplification and we get minus this times this,"},{"Start":"02:19.085 ","End":"02:22.310","Text":"and I\u0027d like to put this back into the denominator so I get"},{"Start":"02:22.310 ","End":"02:26.375","Text":"natural log squared of x over x."},{"Start":"02:26.375 ","End":"02:28.220","Text":"That\u0027s the x to the minus 1."},{"Start":"02:28.220 ","End":"02:31.370","Text":"Minus with minus gives me plus."},{"Start":"02:31.370 ","End":"02:35.300","Text":"But I also like to take the 2 out."},{"Start":"02:35.300 ","End":"02:38.335","Text":"Let\u0027s see what we get. We get plus."},{"Start":"02:38.335 ","End":"02:41.780","Text":"The minus with the minus gives me a plus."},{"Start":"02:41.780 ","End":"02:49.145","Text":"The 2 comes out in front and the x minus 1 with the 1 minus x will combine."},{"Start":"02:49.145 ","End":"02:54.660","Text":"What we\u0027ll get is the 2 here integral."},{"Start":"02:54.660 ","End":"02:57.510","Text":"1 over x, this is also 1 over x,"},{"Start":"02:57.510 ","End":"03:00.050","Text":"together it\u0027s 1 over x squared."},{"Start":"03:00.050 ","End":"03:03.590","Text":"I\u0027ll write this as, or the other way round."},{"Start":"03:03.590 ","End":"03:06.875","Text":"I can write this 1 over x as x to the minus 1,"},{"Start":"03:06.875 ","End":"03:08.555","Text":"which I would prefer to do."},{"Start":"03:08.555 ","End":"03:10.520","Text":"This bit is x to the minus 1."},{"Start":"03:10.520 ","End":"03:14.510","Text":"Then I\u0027ll get natural log of"},{"Start":"03:14.510 ","End":"03:20.510","Text":"x times x to the minus 1 x to the minus 1 is x to the minus 2dx."},{"Start":"03:20.510 ","End":"03:23.585","Text":"I prefer to leave this in exponent form"},{"Start":"03:23.585 ","End":"03:26.735","Text":"because I\u0027m probably going to want to integrate it."},{"Start":"03:26.735 ","End":"03:32.390","Text":"We\u0027re going to do this bit separately also by parts."},{"Start":"03:32.390 ","End":"03:33.865","Text":"Let me just highlight it."},{"Start":"03:33.865 ","End":"03:38.045","Text":"I\u0027m going to do this bit at the side and then we\u0027ll come back here."},{"Start":"03:38.045 ","End":"03:48.115","Text":"What I want is the integral of natural log of x times x to the minus 2dx."},{"Start":"03:48.115 ","End":"03:51.285","Text":"Again, we\u0027re going to do it by parts."},{"Start":"03:51.285 ","End":"03:53.895","Text":"We have the formula still here."},{"Start":"03:53.895 ","End":"03:55.635","Text":"This we\u0027re going to want to differentiate,"},{"Start":"03:55.635 ","End":"03:59.795","Text":"this bit will be u and this bit will be dv."},{"Start":"03:59.795 ","End":"04:03.660","Text":"What we will want here is uv,"},{"Start":"04:03.760 ","End":"04:06.830","Text":"I\u0027m just looking at this formula here,"},{"Start":"04:06.830 ","End":"04:11.845","Text":"minus the integral of vdu."},{"Start":"04:11.845 ","End":"04:16.115","Text":"Okay, u we have natural log of x."},{"Start":"04:16.115 ","End":"04:18.730","Text":"If dv is x to the minus 2dx,"},{"Start":"04:18.730 ","End":"04:20.360","Text":"well, we\u0027ve done this before."},{"Start":"04:20.360 ","End":"04:24.020","Text":"We just get minus x to the minus 1."},{"Start":"04:24.020 ","End":"04:30.965","Text":"Again minus x to the minus 1 and du is just the derivative of natural log of x,"},{"Start":"04:30.965 ","End":"04:32.990","Text":"which is 1 over x,"},{"Start":"04:32.990 ","End":"04:35.660","Text":"which I could also write as x to the minus 1"},{"Start":"04:35.660 ","End":"04:39.380","Text":"we\u0027ll see which way we want to do it, and dx."},{"Start":"04:39.380 ","End":"04:44.410","Text":"Once again, we have integration by parts."},{"Start":"04:44.410 ","End":"04:47.255","Text":"Yeah. We have another integral to do."},{"Start":"04:47.255 ","End":"04:49.070","Text":"Let\u0027s just simplify it."},{"Start":"04:49.070 ","End":"04:55.190","Text":"Once again, I\u0027ll put this on the denominator so minus natural log of x over x."},{"Start":"04:55.190 ","End":"04:59.025","Text":"Minus with minus is plus,"},{"Start":"04:59.025 ","End":"05:01.215","Text":"we did this thing before."},{"Start":"05:01.215 ","End":"05:04.350","Text":"This I\u0027ll write as x to the minus 1 like I did before."},{"Start":"05:04.350 ","End":"05:09.720","Text":"We get integral of x to the minus 2dx."},{"Start":"05:11.210 ","End":"05:15.830","Text":"Again, we\u0027re getting the integral of x to the minus 2,"},{"Start":"05:15.830 ","End":"05:20.990","Text":"we\u0027ve already determined is minus x to the minus 1."},{"Start":"05:20.990 ","End":"05:25.205","Text":"Minus natural log of x over x."},{"Start":"05:25.205 ","End":"05:28.415","Text":"This minus x to the minus 1,"},{"Start":"05:28.415 ","End":"05:30.455","Text":"just like we had before,"},{"Start":"05:30.455 ","End":"05:32.480","Text":"minus x to the minus 1."},{"Start":"05:32.480 ","End":"05:35.300","Text":"This thing here is the same as this thing here."},{"Start":"05:35.300 ","End":"05:37.310","Text":"Minus x to the minus 1."},{"Start":"05:37.310 ","End":"05:42.650","Text":"I\u0027ll write it as 1 over x this time because I don\u0027t want the negative powers anymore."},{"Start":"05:42.650 ","End":"05:47.225","Text":"But we\u0027re not done here because this is just the highlighted part."},{"Start":"05:47.225 ","End":"05:49.555","Text":"Maybe I\u0027m going to go back to,"},{"Start":"05:49.555 ","End":"05:51.195","Text":"maybe I\u0027ll highlight this too."},{"Start":"05:51.195 ","End":"05:54.765","Text":"Now we\u0027re going to go back up to this line here."},{"Start":"05:54.765 ","End":"05:57.980","Text":"What we finally get is"},{"Start":"05:57.980 ","End":"06:08.285","Text":"this bit minus natural log squared of x over x plus twice the highlighted bit,"},{"Start":"06:08.285 ","End":"06:10.225","Text":"which I can just copy from here,"},{"Start":"06:10.225 ","End":"06:15.435","Text":"minus log x over x minus 1 over x,"},{"Start":"06:15.435 ","End":"06:19.690","Text":"and finally plus C. That\u0027s it."}],"ID":4431},{"Watched":false,"Name":"Exercise 19","Duration":"7m 14s","ChapterTopicVideoID":4421,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.810","Text":"In this exercise, we have to compute the integral of e to the x cosine x dx."},{"Start":"00:06.810 ","End":"00:09.330","Text":"As you\u0027ll see later as a bonus,"},{"Start":"00:09.330 ","End":"00:14.580","Text":"we\u0027ll also get the integral of e to the x times sine x. I want to do"},{"Start":"00:14.580 ","End":"00:20.135","Text":"this by parts and I want to remind you the formula for integration by parts."},{"Start":"00:20.135 ","End":"00:29.535","Text":"That is that the integral of udv is equal to uv minus the integral of vdu."},{"Start":"00:29.535 ","End":"00:33.465","Text":"The question is, which is u and which is dv?"},{"Start":"00:33.465 ","End":"00:35.120","Text":"Turns out it doesn\u0027t really matter,"},{"Start":"00:35.120 ","End":"00:40.025","Text":"because we can easily integrate or differentiate e to the x and also cosine x."},{"Start":"00:40.025 ","End":"00:45.845","Text":"But I\u0027ll take this as my u and I\u0027ll take this as my dv,"},{"Start":"00:45.845 ","End":"00:48.740","Text":"but it would work the other way around."},{"Start":"00:48.740 ","End":"00:53.755","Text":"Using the formula, I\u0027m going to get uv."},{"Start":"00:53.755 ","End":"00:57.560","Text":"First of all, u, which is e to the x,"},{"Start":"00:57.560 ","End":"01:00.830","Text":"and now v. What is v?"},{"Start":"01:00.830 ","End":"01:02.975","Text":"If dv is cosine x,"},{"Start":"01:02.975 ","End":"01:05.750","Text":"v is just the antiderivative,"},{"Start":"01:05.750 ","End":"01:08.720","Text":"the integral of cosine x, which is sine x."},{"Start":"01:08.720 ","End":"01:12.715","Text":"That\u0027s going to be my v sine x."},{"Start":"01:12.715 ","End":"01:19.220","Text":"That\u0027s my u, that\u0027s my v minus the integral of v again,"},{"Start":"01:19.220 ","End":"01:24.065","Text":"which is sine x. I need here du,"},{"Start":"01:24.065 ","End":"01:27.904","Text":"du is just the derivative of e to the x,"},{"Start":"01:27.904 ","End":"01:31.980","Text":"so it\u0027s e to the x dx."},{"Start":"01:32.290 ","End":"01:35.920","Text":"Let me make things sharper,"},{"Start":"01:35.920 ","End":"01:38.700","Text":"remove the u and v stuff."},{"Start":"01:38.700 ","End":"01:42.280","Text":"But what we see here is discouraging,"},{"Start":"01:42.280 ","End":"01:46.040","Text":"because we have the integral of e to the x cosine x and we"},{"Start":"01:46.040 ","End":"01:50.255","Text":"have to compute the integral of e to the x sine x."},{"Start":"01:50.255 ","End":"01:53.060","Text":"Who says this is any easier than this,"},{"Start":"01:53.060 ","End":"01:57.185","Text":"but bear with me and you\u0027ll see that there\u0027s a way out."},{"Start":"01:57.185 ","End":"02:00.545","Text":"What we\u0027re going to do is at the side,"},{"Start":"02:00.545 ","End":"02:05.435","Text":"compute the integral of e to the x sine x."},{"Start":"02:05.435 ","End":"02:13.640","Text":"The integral of e to the x sine x dx,"},{"Start":"02:13.640 ","End":"02:15.210","Text":"which is what this is here,"},{"Start":"02:15.210 ","End":"02:16.440","Text":"I just reversed the order,"},{"Start":"02:16.440 ","End":"02:18.350","Text":"so it looks like this is equal to,"},{"Start":"02:18.350 ","End":"02:20.945","Text":"and we\u0027re going to do it by parts again."},{"Start":"02:20.945 ","End":"02:24.110","Text":"But the important thing is to make the same choice."},{"Start":"02:24.110 ","End":"02:29.225","Text":"If the exponential was u and the trigonometric was dv,"},{"Start":"02:29.225 ","End":"02:32.270","Text":"then it should be the same here and vice versa."},{"Start":"02:32.270 ","End":"02:35.620","Text":"I\u0027m not going to write all the u and v. I\u0027ll just write it here,"},{"Start":"02:35.620 ","End":"02:38.090","Text":"that this is again u and this is dv."},{"Start":"02:38.090 ","End":"02:41.435","Text":"But if you started the other way around then it should be the other way round."},{"Start":"02:41.435 ","End":"02:46.590","Text":"Very similarly, we\u0027ll get u is e to the x,"},{"Start":"02:46.590 ","End":"02:51.530","Text":"v this time will be minus cosine x because it\u0027s the anti-derivative of sine"},{"Start":"02:51.530 ","End":"02:56.910","Text":"x minus cosine x minus the integral."},{"Start":"02:56.910 ","End":"03:02.190","Text":"Again, v minus cosine x and du, again,"},{"Start":"03:02.190 ","End":"03:05.010","Text":"e to the x dx,"},{"Start":"03:05.010 ","End":"03:11.520","Text":"which equals minus e to the x cosine x."},{"Start":"03:11.520 ","End":"03:13.995","Text":"Minus with minus is plus,"},{"Start":"03:13.995 ","End":"03:17.970","Text":"plus the integral of,"},{"Start":"03:17.970 ","End":"03:19.545","Text":"if I reverse the order,"},{"Start":"03:19.545 ","End":"03:24.135","Text":"e to the x cosine x dx."},{"Start":"03:24.135 ","End":"03:27.530","Text":"We\u0027re back where we started because look,"},{"Start":"03:27.530 ","End":"03:33.335","Text":"we started with e to the integral of e to the x cosine x and here we have to do it again,"},{"Start":"03:33.335 ","End":"03:35.645","Text":"but don\u0027t worry, there is a way out."},{"Start":"03:35.645 ","End":"03:40.775","Text":"What I suggest doing is to let this whole thing,"},{"Start":"03:40.775 ","End":"03:42.350","Text":"which is equal to this,"},{"Start":"03:42.350 ","End":"03:45.215","Text":"just plug it in and let\u0027s see what happens."},{"Start":"03:45.215 ","End":"03:50.270","Text":"I\u0027m reading from this line and I\u0027m writing this over here."},{"Start":"03:50.270 ","End":"03:58.400","Text":"What I get is that the integral of e to the x cosine x dx is equal to,"},{"Start":"03:58.400 ","End":"04:06.180","Text":"copying this, e to the x sine x minus all these highlighted bits of the minus,"},{"Start":"04:06.180 ","End":"04:08.345","Text":"we\u0027ll make this a plus and this a minus."},{"Start":"04:08.345 ","End":"04:12.785","Text":"I get plus e to the x cosine x"},{"Start":"04:12.785 ","End":"04:21.225","Text":"minus the integral of e to the x cosine x dx."},{"Start":"04:21.225 ","End":"04:23.415","Text":"Perhaps, you can see the way out."},{"Start":"04:23.415 ","End":"04:25.870","Text":"What I\u0027m going to do is bring this to the other side,"},{"Start":"04:25.870 ","End":"04:27.500","Text":"because this is the same as this."},{"Start":"04:27.500 ","End":"04:36.575","Text":"I get twice integral of e to the x cosine x dx and this is equal to what\u0027s left here,"},{"Start":"04:36.575 ","End":"04:43.130","Text":"e to the x sine x plus e to the x cosine x."},{"Start":"04:43.130 ","End":"04:46.420","Text":"All that remains to be done is to divide by 2."},{"Start":"04:46.420 ","End":"04:49.105","Text":"If I take both sides and divide by 2,"},{"Start":"04:49.105 ","End":"04:52.400","Text":"I\u0027ll get the integral of e to"},{"Start":"04:52.400 ","End":"05:00.965","Text":"the x cosine x dx is simply equal to 1.5 e to the x."},{"Start":"05:00.965 ","End":"05:03.530","Text":"You know what, let me take this outside the brackets."},{"Start":"05:03.530 ","End":"05:09.085","Text":"Sine x plus cosine x."},{"Start":"05:09.085 ","End":"05:13.905","Text":"Very good, and then plus c, that\u0027s the answer."},{"Start":"05:13.905 ","End":"05:17.690","Text":"But I want to continue because we can very"},{"Start":"05:17.690 ","End":"05:21.755","Text":"easily kill 2 birds with 1 stone as the saying goes,"},{"Start":"05:21.755 ","End":"05:27.095","Text":"and figure out what is e to the x, sine x also."},{"Start":"05:27.095 ","End":"05:30.950","Text":"Let\u0027s see now, I\u0027m reading from here now,"},{"Start":"05:30.950 ","End":"05:33.185","Text":"I\u0027m going to use these couple of lines."},{"Start":"05:33.185 ","End":"05:41.720","Text":"Integral of e to the x sine x dx is equal to,"},{"Start":"05:41.720 ","End":"05:44.465","Text":"and I\u0027m going to read from this here,"},{"Start":"05:44.465 ","End":"05:50.345","Text":"is equal to minus e to the x cosine x,"},{"Start":"05:50.345 ","End":"05:56.135","Text":"but not plus this because this we\u0027ve already just computed and that\u0027s this."},{"Start":"05:56.135 ","End":"06:02.945","Text":"What I get is plus 1.5 e to the x, sine"},{"Start":"06:02.945 ","End":"06:11.075","Text":"x plus cosine x. I leave the c out for the moment, we\u0027ll add it at the end."},{"Start":"06:11.075 ","End":"06:12.785","Text":"This is equal to,"},{"Start":"06:12.785 ","End":"06:16.235","Text":"if I take e to the x outside the brackets,"},{"Start":"06:16.235 ","End":"06:23.605","Text":"what I\u0027ll get is minus cosine x plus 0.5 Cosine x is minus 0.5"},{"Start":"06:23.605 ","End":"06:31.710","Text":"Cosine x minus 0.5 Cosine x and here plus 0.5 sine x."},{"Start":"06:31.710 ","End":"06:37.805","Text":"Again, plus c, but let me just write it in a form that is similar to this."},{"Start":"06:37.805 ","End":"06:40.310","Text":"What I get is that,"},{"Start":"06:40.310 ","End":"06:41.930","Text":"and I want to write this as a result,"},{"Start":"06:41.930 ","End":"06:48.420","Text":"the integral of e to the x sine x dx is equal to,"},{"Start":"06:48.420 ","End":"06:52.020","Text":"also, I\u0027ll take the 1.5 e to the x,"},{"Start":"06:52.020 ","End":"06:54.030","Text":"and I prefer to change the order,"},{"Start":"06:54.030 ","End":"07:03.240","Text":"sine x minus cosine x plus c. I\u0027ve highlighted the 2 results,"},{"Start":"07:03.240 ","End":"07:04.850","Text":"notice how similar they are."},{"Start":"07:04.850 ","End":"07:06.620","Text":"The only real difference is that here,"},{"Start":"07:06.620 ","End":"07:10.385","Text":"there is a plus and here there is a minus."},{"Start":"07:10.385 ","End":"07:15.690","Text":"That\u0027s it, we\u0027re done. We killed 2 birds with 1 stone. Very good work."}],"ID":4432},{"Watched":false,"Name":"Exercise 20","Duration":"9m 32s","ChapterTopicVideoID":4422,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"Here we have to compute the integral of e to the 2x sine 4x dx."},{"Start":"00:04.530 ","End":"00:06.420","Text":"I\u0027ve copied it here,"},{"Start":"00:06.420 ","End":"00:07.890","Text":"and I\u0027d like to do it by parts."},{"Start":"00:07.890 ","End":"00:10.800","Text":"I\u0027ll remind you of the formula for integration by parts,"},{"Start":"00:10.800 ","End":"00:15.765","Text":"and that is the integral of udv is equal to"},{"Start":"00:15.765 ","End":"00:21.150","Text":"uv minus the integral of vdu."},{"Start":"00:21.150 ","End":"00:23.295","Text":"The usual question is,"},{"Start":"00:23.295 ","End":"00:25.765","Text":"which is v and which is u?"},{"Start":"00:25.765 ","End":"00:28.130","Text":"In this case, it turns out that both will work."},{"Start":"00:28.130 ","End":"00:31.835","Text":"We had a similar but easier one of these before,"},{"Start":"00:31.835 ","End":"00:34.595","Text":"and really doesn\u0027t matter."},{"Start":"00:34.595 ","End":"00:36.785","Text":"I\u0027ll take the first one,"},{"Start":"00:36.785 ","End":"00:38.810","Text":"this one to be u,"},{"Start":"00:38.810 ","End":"00:41.690","Text":"and I\u0027m going to let this bit be dv."},{"Start":"00:41.690 ","End":"00:45.799","Text":"It would work the other way round too, a bit differently."},{"Start":"00:45.799 ","End":"00:51.125","Text":"Same answer. What I have to do is substitute in this formula."},{"Start":"00:51.125 ","End":"00:53.450","Text":"Let me just prepare a template."},{"Start":"00:53.450 ","End":"00:56.240","Text":"I\u0027m going to need a u, I\u0027m going to need a v,"},{"Start":"00:56.240 ","End":"00:58.190","Text":"I\u0027m going to need a minus the integral,"},{"Start":"00:58.190 ","End":"00:59.705","Text":"I\u0027m going to need v again,"},{"Start":"00:59.705 ","End":"01:02.080","Text":"and I\u0027m going to need du."},{"Start":"01:02.080 ","End":"01:04.145","Text":"Let\u0027s see what we have."},{"Start":"01:04.145 ","End":"01:08.195","Text":"U is e to the 2x, no problem there."},{"Start":"01:08.195 ","End":"01:11.730","Text":"What about v? Here we have sine of 4x."},{"Start":"01:11.730 ","End":"01:22.045","Text":"Well, the integral of sine is minus cosine x. I would like to write minus cosine 4x."},{"Start":"01:22.045 ","End":"01:27.470","Text":"But there\u0027s an internal derivative and it\u0027s a linear."},{"Start":"01:27.630 ","End":"01:30.395","Text":"The inner function is 4x,"},{"Start":"01:30.395 ","End":"01:32.260","Text":"and its derivative is a constant."},{"Start":"01:32.260 ","End":"01:35.710","Text":"In this case, you\u0027re allowed to use the same trick in"},{"Start":"01:35.710 ","End":"01:40.195","Text":"integration as in differentiation only you divide by the internal derivative,"},{"Start":"01:40.195 ","End":"01:42.545","Text":"so I\u0027m dividing by 4."},{"Start":"01:42.545 ","End":"01:49.060","Text":"Now that I have v, I can write it again as minus 1/4 cosine 4x."},{"Start":"01:49.060 ","End":"01:51.560","Text":"Just change the order a bit."},{"Start":"01:52.820 ","End":"01:57.075","Text":"Du I get by integrating this."},{"Start":"01:57.075 ","End":"01:59.820","Text":"Again, there\u0027s an internal derivative which is linear."},{"Start":"01:59.820 ","End":"02:06.429","Text":"It\u0027s e to the 2x times the inner derivative and then dx."},{"Start":"02:06.680 ","End":"02:09.930","Text":"Let\u0027s just slightly simplify,"},{"Start":"02:09.930 ","End":"02:12.315","Text":"get rid of this stuff."},{"Start":"02:12.315 ","End":"02:22.350","Text":"This equals minus 1/4 e to the 2x cosine 4x."},{"Start":"02:22.350 ","End":"02:25.710","Text":"Minus and minus is plus."},{"Start":"02:25.710 ","End":"02:29.775","Text":"2 with 1/4 is 1/2,"},{"Start":"02:29.775 ","End":"02:36.720","Text":"e to the 2x cosine 4x."},{"Start":"02:36.720 ","End":"02:42.620","Text":"I forgot the integral sign which I can put after the 1/2, dx."},{"Start":"02:42.660 ","End":"02:50.639","Text":"What I\u0027d like to do is do just this bit also in parts."},{"Start":"02:50.780 ","End":"02:56.645","Text":"I copied this over here and now we have something similar to the original."},{"Start":"02:56.645 ","End":"02:58.760","Text":"We\u0027re going to do this again by parts."},{"Start":"02:58.760 ","End":"03:02.255","Text":"Now, what\u0027s important is that we make the same choice as before."},{"Start":"03:02.255 ","End":"03:06.430","Text":"Before I remember that we chose this as,"},{"Start":"03:06.430 ","End":"03:14.405","Text":"this bit was our u and this the exponential part and the trigonometric part was dv."},{"Start":"03:14.405 ","End":"03:17.300","Text":"It\u0027s important that here we do the same thing,"},{"Start":"03:17.300 ","End":"03:21.120","Text":"that this will be our u and this will be the dv."},{"Start":"03:21.120 ","End":"03:24.830","Text":"If you did it the other way around and vice versa,"},{"Start":"03:24.830 ","End":"03:27.500","Text":"so very similar."},{"Start":"03:27.500 ","End":"03:29.630","Text":"I\u0027m going to do it quicker this time."},{"Start":"03:29.630 ","End":"03:32.450","Text":"It\u0027s equal to uv."},{"Start":"03:32.450 ","End":"03:37.735","Text":"U is e to the 2x and v,"},{"Start":"03:37.735 ","End":"03:39.800","Text":"just like before, we\u0027ll put the sine,"},{"Start":"03:39.800 ","End":"03:41.660","Text":"we got minus cosine here,"},{"Start":"03:41.660 ","End":"03:44.075","Text":"the integral of cosine is sine."},{"Start":"03:44.075 ","End":"03:47.470","Text":"What we\u0027re going to get is sine"},{"Start":"03:47.470 ","End":"03:53.660","Text":"4x and we\u0027re going to divide by 4 because of the inner derivative,"},{"Start":"03:53.660 ","End":"03:55.415","Text":"which is a constant."},{"Start":"03:55.415 ","End":"03:59.545","Text":"Then minus the integral."},{"Start":"03:59.545 ","End":"04:06.585","Text":"Again, v is going to be sine 4x times 1/4,"},{"Start":"04:06.585 ","End":"04:10.955","Text":"and du is just going to be the same as before."},{"Start":"04:10.955 ","End":"04:15.640","Text":"It\u0027s going to be e to the 2x times 2 dx."},{"Start":"04:15.640 ","End":"04:18.420","Text":"Just simplifying this a bit,"},{"Start":"04:18.420 ","End":"04:28.740","Text":"we get 1/4 e to the 2x sine 4x minus,"},{"Start":"04:28.740 ","End":"04:32.220","Text":"let\u0027s see, 2 and 1/4 together"},{"Start":"04:32.220 ","End":"04:39.390","Text":"give 1/2 and the integral,"},{"Start":"04:39.390 ","End":"04:40.905","Text":"I just switched the order,"},{"Start":"04:40.905 ","End":"04:48.660","Text":"e to the 2x sine 4x dx."},{"Start":"04:48.660 ","End":"04:52.595","Text":"Now, it looks like we\u0027re going round in circles"},{"Start":"04:52.595 ","End":"04:57.725","Text":"because we started off with the integral of e to the 2x sine 4x."},{"Start":"04:57.725 ","End":"05:01.505","Text":"Here we have again the integral of e to the 2x sine 4x."},{"Start":"05:01.505 ","End":"05:06.070","Text":"But bear with me and you\u0027ll see we can find a way out."},{"Start":"05:06.070 ","End":"05:08.885","Text":"I highlighted this in turquoise,"},{"Start":"05:08.885 ","End":"05:12.245","Text":"same as this because it is the same."},{"Start":"05:12.245 ","End":"05:18.710","Text":"What I\u0027m going to do is put this instead of this in this line."},{"Start":"05:18.710 ","End":"05:21.800","Text":"What I\u0027m saying is if we get back to the beginning,"},{"Start":"05:21.800 ","End":"05:31.050","Text":"the integral of e to the 2x sine 4x dx is equal to."},{"Start":"05:31.050 ","End":"05:32.780","Text":"Now, from this line,"},{"Start":"05:32.780 ","End":"05:34.985","Text":"I\u0027m going to just copy this line."},{"Start":"05:34.985 ","End":"05:36.800","Text":"Not exactly copy it,"},{"Start":"05:36.800 ","End":"05:44.315","Text":"but minus 1/4 e to the 2x cosine 4x plus 1/2."},{"Start":"05:44.315 ","End":"05:47.060","Text":"Now is where I do the replacement."},{"Start":"05:47.060 ","End":"05:48.935","Text":"Instead of this integral,"},{"Start":"05:48.935 ","End":"05:51.980","Text":"I\u0027m going to put this expression here."},{"Start":"05:51.980 ","End":"06:01.080","Text":"1/4 e to the 2x sine 4x."},{"Start":"06:01.530 ","End":"06:05.560","Text":"I think I made an error here because I see the"},{"Start":"06:05.560 ","End":"06:08.885","Text":"2 with the 1/4 was a 1/2 and I wrote here 4."},{"Start":"06:08.885 ","End":"06:12.110","Text":"I\u0027ll just write over it. This was 1/2."},{"Start":"06:12.110 ","End":"06:16.425","Text":"Sine 4x minus 1/2,"},{"Start":"06:16.425 ","End":"06:19.020","Text":"the integral of e to"},{"Start":"06:19.020 ","End":"06:27.420","Text":"the 2x sine 4x dx,"},{"Start":"06:27.420 ","End":"06:30.090","Text":"close brackets."},{"Start":"06:30.090 ","End":"06:34.415","Text":"We are going to get out of this mess, don\u0027t you worry."},{"Start":"06:34.415 ","End":"06:38.200","Text":"What we\u0027re going to do, I want to open the brackets,"},{"Start":"06:38.200 ","End":"06:41.350","Text":"but I don\u0027t want to write the whole thing again."},{"Start":"06:41.350 ","End":"06:47.275","Text":"What I\u0027m going to do is I just take the 1/2 with the 1/4 this together."},{"Start":"06:47.275 ","End":"06:51.600","Text":"If I take the 1/2 off and I put it in,"},{"Start":"06:51.600 ","End":"06:53.475","Text":"this will give me 1/8."},{"Start":"06:53.475 ","End":"06:55.470","Text":"The 1/2 with the minus 1/2."},{"Start":"06:55.470 ","End":"06:57.240","Text":"This will give me 1/4."},{"Start":"06:57.240 ","End":"07:01.250","Text":"Then I can just dispense with this 1/2 and with the brackets."},{"Start":"07:01.250 ","End":"07:03.600","Text":"Just save a step."},{"Start":"07:03.680 ","End":"07:09.650","Text":"Now, what I\u0027m going to do is this integral I\u0027m going to put over to the other side."},{"Start":"07:09.650 ","End":"07:14.300","Text":"Now, here I have the integral plus 1/4 of the same thing."},{"Start":"07:14.300 ","End":"07:16.790","Text":"Now, in general in algebra,"},{"Start":"07:16.790 ","End":"07:19.430","Text":"if I have something like, I don\u0027t know,"},{"Start":"07:19.430 ","End":"07:22.940","Text":"a and I add another 1/4a,"},{"Start":"07:22.940 ","End":"07:26.430","Text":"whatever that is, that\u0027s going to be 5/4a."},{"Start":"07:26.440 ","End":"07:28.910","Text":"The same thing with this integral."},{"Start":"07:28.910 ","End":"07:30.235","Text":"When I bring this over,"},{"Start":"07:30.235 ","End":"07:35.940","Text":"I\u0027m going to get 5/4 of the integral of e"},{"Start":"07:35.940 ","End":"07:42.450","Text":"to the 2x sine 4x dx is equal to,"},{"Start":"07:42.450 ","End":"07:45.285","Text":"and then there\u0027s these first 2 terms."},{"Start":"07:45.285 ","End":"07:48.735","Text":"I\u0027ll take e to the 2x outside the brackets."},{"Start":"07:48.735 ","End":"07:53.415","Text":"I have e to the 2x and then I have,"},{"Start":"07:53.415 ","End":"07:58.420","Text":"there\u0027s 1/4 and 1/4, so let me take the 1/4 also outside the brackets."},{"Start":"07:58.420 ","End":"08:08.665","Text":"What I\u0027m left with is minus cosine 4x and here I have 1/8."},{"Start":"08:08.665 ","End":"08:10.895","Text":"You know what? I changed my mind."},{"Start":"08:10.895 ","End":"08:13.145","Text":"I don\u0027t want to take the 1/4 out."},{"Start":"08:13.145 ","End":"08:15.740","Text":"Put the 1 minus 1/4 back in."},{"Start":"08:15.740 ","End":"08:17.975","Text":"I\u0027m just taking e to the 2x out."},{"Start":"08:17.975 ","End":"08:28.360","Text":"From here, I\u0027m left with plus 1/8 e to the 2x I took out and sine 4x."},{"Start":"08:28.760 ","End":"08:36.799","Text":"Now, all I have to do is divide both sides by 5 over 4."},{"Start":"08:36.799 ","End":"08:40.550","Text":"I\u0027ll get my final answer that the"},{"Start":"08:40.550 ","End":"08:47.600","Text":"integral of e to the 2x sine 4x dx,"},{"Start":"08:47.600 ","End":"08:51.480","Text":"finally is just equal to 4/5."},{"Start":"08:51.520 ","End":"08:54.350","Text":"I\u0027m just copying this again."},{"Start":"08:54.350 ","End":"09:00.430","Text":"E to the 2x times minus 1/4"},{"Start":"09:00.430 ","End":"09:08.370","Text":"cosine 4x plus 1/8 sine 4x."},{"Start":"09:08.370 ","End":"09:11.190","Text":"I think I\u0027m missing an s here."},{"Start":"09:11.190 ","End":"09:13.750","Text":"Then plus C at the very end."},{"Start":"09:13.750 ","End":"09:15.320","Text":"I should have put it maybe before,"},{"Start":"09:15.320 ","End":"09:17.765","Text":"but I\u0027m going to have to put it once at the end."},{"Start":"09:17.765 ","End":"09:25.425","Text":"But those are ones it can be simplified and 4/5 times 1/4 is 1/5 and so on."},{"Start":"09:25.425 ","End":"09:26.960","Text":"You could combine the fractions,"},{"Start":"09:26.960 ","End":"09:29.645","Text":"you could multiply, but I\u0027m not going to do that."},{"Start":"09:29.645 ","End":"09:33.270","Text":"We\u0027re going to leave this as our answer, and we\u0027re done."}],"ID":4433},{"Watched":false,"Name":"Exercise 21","Duration":"7m 59s","ChapterTopicVideoID":4423,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"We have to compute the following integral."},{"Start":"00:02.310 ","End":"00:03.960","Text":"I\u0027ve copied it over here."},{"Start":"00:03.960 ","End":"00:07.005","Text":"Note that there\u0027s a 1 minus x squared here."},{"Start":"00:07.005 ","End":"00:13.155","Text":"The normal way to do it or the usual way is the trigonometric substitution were I let x"},{"Start":"00:13.155 ","End":"00:19.245","Text":"equals sine t or cosine t. But we\u0027re practicing integration by parts,"},{"Start":"00:19.245 ","End":"00:20.580","Text":"and that\u0027s how we\u0027re going to do it,"},{"Start":"00:20.580 ","End":"00:25.155","Text":"and here\u0027s the formula for integration by parts which I\u0027ve prepared."},{"Start":"00:25.155 ","End":"00:32.760","Text":"Clearly this one has to be my u and this will be the dv,"},{"Start":"00:32.760 ","End":"00:34.985","Text":"and so according to the formula,"},{"Start":"00:34.985 ","End":"00:39.415","Text":"what I need is u v,"},{"Start":"00:39.415 ","End":"00:41.820","Text":"this will be my u, my v here,"},{"Start":"00:41.820 ","End":"00:44.520","Text":"I\u0027m putting place holders minus the integral."},{"Start":"00:44.520 ","End":"00:50.215","Text":"Here I\u0027m going to have v and here I\u0027m going to have du,"},{"Start":"00:50.215 ","End":"00:53.775","Text":"so let\u0027s see, here we have already,"},{"Start":"00:53.775 ","End":"00:59.840","Text":"is square root of 1 minus x squared."},{"Start":"00:59.840 ","End":"01:02.150","Text":"If dv is dx,"},{"Start":"01:02.150 ","End":"01:07.830","Text":"then v is just x here and here,"},{"Start":"01:07.830 ","End":"01:12.775","Text":"and du would be the derivative of this."},{"Start":"01:12.775 ","End":"01:15.410","Text":"If it\u0027s a square root,"},{"Start":"01:15.410 ","End":"01:23.660","Text":"it would be 1 over twice the square root of 1 minus x squared."},{"Start":"01:23.660 ","End":"01:28.835","Text":"But there\u0027s a matter of an internal derivative of minus 2x,"},{"Start":"01:28.835 ","End":"01:34.915","Text":"so I\u0027ll just change this 1 to minus 2x,"},{"Start":"01:34.915 ","End":"01:39.120","Text":"there, and let\u0027s not forget the dx."},{"Start":"01:39.120 ","End":"01:41.845","Text":"I\u0027d like to get rid of this clutter."},{"Start":"01:41.845 ","End":"01:44.060","Text":"I could simplify this a bit."},{"Start":"01:44.060 ","End":"01:51.245","Text":"For example, this minus will cancel with this minus and the 2,"},{"Start":"01:51.245 ","End":"01:53.970","Text":"will go with the 2."},{"Start":"01:58.850 ","End":"02:03.710","Text":"Also notice that this x and this x together will give x squared,"},{"Start":"02:03.710 ","End":"02:10.114","Text":"so we end up with x square root of 1 minus x squared,"},{"Start":"02:10.114 ","End":"02:15.770","Text":"plus the integral of x squared"},{"Start":"02:15.770 ","End":"02:22.805","Text":"over square root of 1 minus x squared, dx."},{"Start":"02:22.805 ","End":"02:25.205","Text":"I want do this bit at the side,"},{"Start":"02:25.205 ","End":"02:27.500","Text":"the bit I\u0027ve highlighted, so let\u0027s see."},{"Start":"02:27.500 ","End":"02:35.045","Text":"We have the integral of x squared over the square root of 1 minus x squared dx."},{"Start":"02:35.045 ","End":"02:36.830","Text":"This is a little bit tricky,"},{"Start":"02:36.830 ","End":"02:41.825","Text":"but actually quite easy once you\u0027re used to the algebraic tricks we do."},{"Start":"02:41.825 ","End":"02:46.055","Text":"If I had on the numerator 1 minus x squared,"},{"Start":"02:46.055 ","End":"02:47.270","Text":"that would be great,"},{"Start":"02:47.270 ","End":"02:52.160","Text":"because then I\u0027d have a over the square root of a and that\u0027s easily simplifiable."},{"Start":"02:52.160 ","End":"02:55.400","Text":"What I have in the numerator is not 1 minus x squared,"},{"Start":"02:55.400 ","End":"02:58.110","Text":"but it\u0027s not that far from it."},{"Start":"03:03.050 ","End":"03:07.560","Text":"What I can say is, and let me do this at the side here."},{"Start":"03:09.860 ","End":"03:13.440","Text":"I\u0027d like it to be 1 minus x squared,"},{"Start":"03:13.440 ","End":"03:18.875","Text":"so how am I going to alter 1 minus x squared to be x squared?"},{"Start":"03:18.875 ","End":"03:22.685","Text":"Well, for the first thing I could do would be to write a minus."},{"Start":"03:22.685 ","End":"03:26.405","Text":"If I wrote minus of 1 minus x squared,"},{"Start":"03:26.405 ","End":"03:33.520","Text":"that would be closer because that would be minus 1 plus x squared."},{"Start":"03:33.520 ","End":"03:36.530","Text":"Now, if this is minus 1 plus x squared,"},{"Start":"03:36.530 ","End":"03:39.690","Text":"all I\u0027d have to do then is to add 1."},{"Start":"03:40.480 ","End":"03:45.020","Text":"If you open it up you\u0027ll see I have minus 1 plus x"},{"Start":"03:45.020 ","End":"03:50.340","Text":"squared plus 1 and the minus 1 and plus 1 cancel, so that will do it."},{"Start":"03:50.340 ","End":"03:55.910","Text":"All I have to do is because this equals this,"},{"Start":"03:55.910 ","End":"04:04.865","Text":"I can now go back here and rewrite it just with this algebra as the integral."},{"Start":"04:04.865 ","End":"04:06.500","Text":"Instead of x squared,"},{"Start":"04:06.500 ","End":"04:08.660","Text":"I\u0027m going to put this and I\u0027ll change the order,"},{"Start":"04:08.660 ","End":"04:18.930","Text":"I\u0027ll put it as 1 minus 1 minus x squared over the same denominator dx."},{"Start":"04:18.930 ","End":"04:23.465","Text":"Now, at this point what I can do is"},{"Start":"04:23.465 ","End":"04:28.760","Text":"to separate it into 2 integrals using this minus sign,"},{"Start":"04:28.760 ","End":"04:33.675","Text":"and so what I get is equal to."},{"Start":"04:33.675 ","End":"04:43.620","Text":"I have the integral of 1 over square root of 1 minus x squared dx,"},{"Start":"04:43.620 ","End":"04:49.565","Text":"minus the integral of this over this."},{"Start":"04:49.565 ","End":"04:53.030","Text":"Now also I want to again a little bit of algebra at the side."},{"Start":"04:53.030 ","End":"04:56.360","Text":"If I let this 1 minus x squared be a,"},{"Start":"04:56.360 ","End":"05:00.920","Text":"what I have here is a over the square root of a,"},{"Start":"05:00.920 ","End":"05:04.205","Text":"and that\u0027s just equal to the square root of a."},{"Start":"05:04.205 ","End":"05:09.725","Text":"All in all, what I have for the second bit is just minus"},{"Start":"05:09.725 ","End":"05:16.995","Text":"the integral of the square root of 1 minus x squared dx."},{"Start":"05:16.995 ","End":"05:20.970","Text":"I\u0027ve broken this integral up into 2 integrals,"},{"Start":"05:20.970 ","End":"05:25.340","Text":"and hopefully these will be easier to solve,"},{"Start":"05:25.340 ","End":"05:26.790","Text":"with that we can solve them."},{"Start":"05:26.790 ","End":"05:31.510","Text":"Well this one I know it\u0027s an immediate integral as you\u0027ll see with the trigonometric one,"},{"Start":"05:31.510 ","End":"05:33.635","Text":"and this we shall see."},{"Start":"05:33.635 ","End":"05:36.710","Text":"As a matter of fact, I don\u0027t know how to do this integral"},{"Start":"05:36.710 ","End":"05:39.760","Text":"because this is exactly the integral I started out with,"},{"Start":"05:39.760 ","End":"05:41.810","Text":"and it appears we\u0027re going round in a circle,"},{"Start":"05:41.810 ","End":"05:46.400","Text":"but you\u0027ve seen the tricks we do before we bring it over to the other side."},{"Start":"05:46.400 ","End":"05:49.040","Text":"Just bear with me and we\u0027ll see."},{"Start":"05:49.040 ","End":"05:54.579","Text":"What we\u0027ve meanwhile done is expanded this highlighted one into this."},{"Start":"05:54.579 ","End":"06:01.430","Text":"What I\u0027m going to do now is just replace this highlighted bit with this."},{"Start":"06:01.430 ","End":"06:03.545","Text":"We start off with this,"},{"Start":"06:03.545 ","End":"06:10.890","Text":"the integral of square root of 1 minus x squared dx equals,"},{"Start":"06:10.890 ","End":"06:14.015","Text":"and now I\u0027m reading off this line here,"},{"Start":"06:14.015 ","End":"06:19.850","Text":"is x square root of 1 minus x squared"},{"Start":"06:19.850 ","End":"06:28.890","Text":"plus the blue stuff."},{"Start":"06:28.890 ","End":"06:35.270","Text":"I forgot to just mention that this bit here there\u0027s an immediate"},{"Start":"06:35.270 ","End":"06:37.010","Text":"integral and it\u0027s actually"},{"Start":"06:37.010 ","End":"06:42.695","Text":"the arc sine of x. I should have done it in another row, never mind."},{"Start":"06:42.695 ","End":"06:48.695","Text":"We get plus arc sine of"},{"Start":"06:48.695 ","End":"06:56.105","Text":"x minus the integral of the square root of 1 minus x squared dx,"},{"Start":"06:56.105 ","End":"06:59.135","Text":"the last bit being the thing that we\u0027re looking for."},{"Start":"06:59.135 ","End":"07:03.920","Text":"But actually this is not bad because this appears with a minus here,"},{"Start":"07:03.920 ","End":"07:08.405","Text":"so if I take this integral and bring it over to the other side,"},{"Start":"07:08.405 ","End":"07:11.055","Text":"and I\u0027ll have twice this integral."},{"Start":"07:11.055 ","End":"07:13.515","Text":"Because I\u0027ll have this thing plus itself,"},{"Start":"07:13.515 ","End":"07:18.570","Text":"so twice the integral of the square root"},{"Start":"07:18.570 ","End":"07:24.445","Text":"of 1 minus x squared dx is just equal to this,"},{"Start":"07:24.445 ","End":"07:32.295","Text":"so it\u0027s x root 1 minus x squared plus arc sine x,"},{"Start":"07:32.295 ","End":"07:33.735","Text":"could add a constant,"},{"Start":"07:33.735 ","End":"07:35.325","Text":"I\u0027ll just could do 1 at the end,"},{"Start":"07:35.325 ","End":"07:37.305","Text":"and then I divide by 2,"},{"Start":"07:37.305 ","End":"07:45.135","Text":"so ultimately our integral of 1 minus x squared dx"},{"Start":"07:45.135 ","End":"07:49.705","Text":"is simply 1/2 of what\u0027s written above,"},{"Start":"07:49.705 ","End":"07:52.840","Text":"x square root of 1 minus x squared,"},{"Start":"07:52.840 ","End":"08:00.070","Text":"plus arc sine of x plus a constant."}],"ID":4434},{"Watched":false,"Name":"Exercise 22","Duration":"5m 27s","ChapterTopicVideoID":4424,"CourseChapterTopicPlaylistID":3684,"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":"Here we have to compute the following integral,"},{"Start":"00:02.430 ","End":"00:04.575","Text":"which I\u0027ve rewritten here."},{"Start":"00:04.575 ","End":"00:06.570","Text":"I\u0027m going to do it by parts,"},{"Start":"00:06.570 ","End":"00:10.185","Text":"that\u0027s why I wrote down the formula for integration by parts."},{"Start":"00:10.185 ","End":"00:12.300","Text":"But in integration by parts we like"},{"Start":"00:12.300 ","End":"00:16.725","Text":"products not quotients so we\u0027re going to have to split this up into a product."},{"Start":"00:16.725 ","End":"00:21.150","Text":"It could be done in several ways but the way that\u0027s going to work"},{"Start":"00:21.150 ","End":"00:26.070","Text":"is to take 1 bit as xe^x,"},{"Start":"00:26.070 ","End":"00:32.969","Text":"and the other bit as 1 over x plus 1 squared dx."},{"Start":"00:32.969 ","End":"00:34.815","Text":"Now I have a product."},{"Start":"00:34.815 ","End":"00:38.150","Text":"Again, we have to choose which is u and which is dv."},{"Start":"00:38.150 ","End":"00:44.645","Text":"The way that\u0027s going to work is if we make this u and this dv."},{"Start":"00:44.645 ","End":"00:48.575","Text":"But perhaps to make things easier,"},{"Start":"00:48.575 ","End":"00:52.280","Text":"I\u0027ll rewrite this as x plus 1^minus 2."},{"Start":"00:52.280 ","End":"00:55.555","Text":"I\u0027ll just erase this and write a negative power,"},{"Start":"00:55.555 ","End":"00:58.290","Text":"it\u0027s just a bit more convenient."},{"Start":"00:58.290 ","End":"01:00.440","Text":"Now according to the formula,"},{"Start":"01:00.440 ","End":"01:03.605","Text":"what I\u0027m going to want is uv,"},{"Start":"01:03.605 ","End":"01:05.900","Text":"so I\u0027m just going to put placeholders here."},{"Start":"01:05.900 ","End":"01:10.380","Text":"I\u0027m going to have u, and then here I\u0027m going to put v. Then I\u0027m"},{"Start":"01:10.380 ","End":"01:15.105","Text":"going to have an integral and in the integral I\u0027m going to have vdu."},{"Start":"01:15.105 ","End":"01:19.270","Text":"Here I\u0027m going to have v and this part\u0027s going to be du."},{"Start":"01:19.270 ","End":"01:23.220","Text":"Let\u0027s see, u we have already is xe^x."},{"Start":"01:23.220 ","End":"01:26.985","Text":"V, we don\u0027t have, we have dv."},{"Start":"01:26.985 ","End":"01:29.130","Text":"We just integrate this."},{"Start":"01:29.130 ","End":"01:33.740","Text":"Notice that there\u0027s an inner function x plus 1 within a derivative 1,"},{"Start":"01:33.740 ","End":"01:35.945","Text":"so it\u0027s treated like x."},{"Start":"01:35.945 ","End":"01:39.275","Text":"We just do the usual as if it was x to the minus 2."},{"Start":"01:39.275 ","End":"01:44.490","Text":"We make it to the minus 1 and 0 over minus 1."},{"Start":"01:44.490 ","End":"01:48.390","Text":"Here again we have v, just copy it,"},{"Start":"01:48.390 ","End":"01:58.385","Text":"x plus 1 to the minus 1 over minus 1 and du is just the derivative of this."},{"Start":"01:58.385 ","End":"02:03.575","Text":"I\u0027m sure you all know the product rule already because here we have a product."},{"Start":"02:03.575 ","End":"02:06.515","Text":"At the end we\u0027re going to have dx."},{"Start":"02:06.515 ","End":"02:08.915","Text":"But before that we\u0027re going to have a product of,"},{"Start":"02:08.915 ","End":"02:14.720","Text":"let\u0027s say the first one as is times the derivative of the second."},{"Start":"02:14.720 ","End":"02:16.490","Text":"Then the other way round,"},{"Start":"02:16.490 ","End":"02:21.320","Text":"the derivative of the first and the second as is."},{"Start":"02:21.320 ","End":"02:23.240","Text":"This is what we have."},{"Start":"02:23.240 ","End":"02:26.525","Text":"I\u0027d like to get rid of the clutter here,"},{"Start":"02:26.525 ","End":"02:30.350","Text":"that\u0027s better and now a bit of simplifying."},{"Start":"02:30.350 ","End":"02:32.920","Text":"Let\u0027s look at this bit here."},{"Start":"02:32.920 ","End":"02:36.050","Text":"What we get, the minus 1 in the denominator can"},{"Start":"02:36.050 ","End":"02:39.185","Text":"be in the numerator and it just comes out as a minus."},{"Start":"02:39.185 ","End":"02:42.530","Text":"Here I have xe^x,"},{"Start":"02:42.530 ","End":"02:44.330","Text":"and this thing to the minus 1,"},{"Start":"02:44.330 ","End":"02:46.570","Text":"I can just stick in the denominator."},{"Start":"02:46.570 ","End":"02:48.995","Text":"This is what I get here."},{"Start":"02:48.995 ","End":"02:52.070","Text":"I should have put some brackets here,"},{"Start":"02:52.070 ","End":"02:56.640","Text":"of course, because it\u0027s the integral of the whole thing."},{"Start":"02:56.720 ","End":"03:02.705","Text":"This minus 1 will cancel with this minus and make it plus."},{"Start":"03:02.705 ","End":"03:08.810","Text":"What I have here is xe^x"},{"Start":"03:08.810 ","End":"03:15.840","Text":"plus e^x over x plus 1 to the minus 1,"},{"Start":"03:15.840 ","End":"03:20.345","Text":"like before, I can just put in the denominator so it\u0027s x plus 1."},{"Start":"03:20.345 ","End":"03:24.390","Text":"Now look, I\u0027m going to do something at the side here."},{"Start":"03:25.450 ","End":"03:32.599","Text":"Xe^x plus e^x is just equal to if I take e^x outside the brackets,"},{"Start":"03:32.599 ","End":"03:34.880","Text":"it will be x plus 1."},{"Start":"03:34.880 ","End":"03:37.470","Text":"Now if I do that here,"},{"Start":"03:38.210 ","End":"03:43.115","Text":"I put this here and this x plus 1 would cancel with this x plus 1,"},{"Start":"03:43.115 ","End":"03:47.165","Text":"and what I\u0027d be left with is just e^x,"},{"Start":"03:47.165 ","End":"03:51.680","Text":"so I can now rewrite the whole thing as"},{"Start":"03:51.680 ","End":"04:02.630","Text":"minus xe^x over x plus 1 plus the integral of just e^x dx."},{"Start":"04:02.630 ","End":"04:05.615","Text":"Now this is an immediate integral."},{"Start":"04:05.615 ","End":"04:15.020","Text":"We know what it is, it\u0027s just e^x itself so I now get minus xe^x over x plus 1,"},{"Start":"04:15.020 ","End":"04:18.710","Text":"which I\u0027m dragging along everywhere, plus e^x."},{"Start":"04:18.710 ","End":"04:23.495","Text":"But now I need the constant of integration and we\u0027re done."},{"Start":"04:23.495 ","End":"04:27.730","Text":"Now we are done but I would like to simplify this,"},{"Start":"04:27.730 ","End":"04:30.040","Text":"so if you don\u0027t feel like the simplification,"},{"Start":"04:30.040 ","End":"04:31.570","Text":"you can just leave now."},{"Start":"04:31.570 ","End":"04:34.930","Text":"But if you do, this will actually turn out much simpler,"},{"Start":"04:34.930 ","End":"04:38.975","Text":"because I can take e^x outside the brackets,"},{"Start":"04:38.975 ","End":"04:41.140","Text":"let\u0027s leave the C out of it for the moment,"},{"Start":"04:41.140 ","End":"04:50.559","Text":"I\u0027ve got e^x and I\u0027ve got minus x over x plus 1 plus 1."},{"Start":"04:50.559 ","End":"04:54.880","Text":"Continuing, if I put a common denominator for this,"},{"Start":"04:54.880 ","End":"04:57.910","Text":"I put it all over x plus 1,"},{"Start":"04:57.910 ","End":"05:04.420","Text":"then here I have minus x and 1 is just x plus 1 over x plus 1."},{"Start":"05:04.420 ","End":"05:11.930","Text":"Then I can say that minus x with plus x cancels and"},{"Start":"05:11.930 ","End":"05:20.770","Text":"ultimately what I\u0027m left with as a final answer is e^x over x plus 1."},{"Start":"05:20.770 ","End":"05:28.440","Text":"Now I\u0027ll just put C in and this is our simplified answer, really done now."}],"ID":4435},{"Watched":false,"Name":"Exercise 23","Duration":"7m 1s","ChapterTopicVideoID":4425,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.715","Text":"Here\u0027s another integral we have to compute,"},{"Start":"00:02.715 ","End":"00:04.710","Text":"and we\u0027re going to do it by parts."},{"Start":"00:04.710 ","End":"00:12.840","Text":"Remember the formula, the integral of udv is equal to uv minus the integral of vdu."},{"Start":"00:12.840 ","End":"00:15.900","Text":"As usual, we have to decide which is u and which is dv."},{"Start":"00:15.900 ","End":"00:19.260","Text":"Turns out, contrary to expectations,"},{"Start":"00:19.260 ","End":"00:22.665","Text":"that actually this is better off as u,"},{"Start":"00:22.665 ","End":"00:26.925","Text":"and this is what we will have as dv."},{"Start":"00:26.925 ","End":"00:30.645","Text":"On the other side, what we\u0027ll need is,"},{"Start":"00:30.645 ","End":"00:32.460","Text":"I\u0027ll just put placeholders,"},{"Start":"00:32.460 ","End":"00:40.590","Text":"is we\u0027ll need u and then v and then minus the integral,"},{"Start":"00:40.590 ","End":"00:42.780","Text":"and here we\u0027ll have v,"},{"Start":"00:42.780 ","End":"00:45.810","Text":"and here we\u0027ll have du."},{"Start":"00:45.810 ","End":"00:53.685","Text":"u we already have is x. v we don\u0027t know."},{"Start":"00:53.685 ","End":"01:00.110","Text":"We need the integral of tangent squared x. I\u0027ll leave"},{"Start":"01:00.110 ","End":"01:06.920","Text":"that for the moment and du we do have because if u is x,"},{"Start":"01:06.920 ","End":"01:09.095","Text":"then du is just dx."},{"Start":"01:09.095 ","End":"01:13.450","Text":"But what we\u0027re missing is what is v equal to?"},{"Start":"01:13.450 ","End":"01:15.330","Text":"That\u0027s the big question."},{"Start":"01:15.330 ","End":"01:18.455","Text":"I suggest we\u0027ll do this as a side exercise."},{"Start":"01:18.455 ","End":"01:20.720","Text":"We know that dv is this,"},{"Start":"01:20.720 ","End":"01:24.320","Text":"so v is just the integral of tangent squared x."},{"Start":"01:24.320 ","End":"01:26.330","Text":"Let\u0027s do a side exercise."},{"Start":"01:26.330 ","End":"01:28.890","Text":"You know what, I\u0027ll highlight it."},{"Start":"01:29.900 ","End":"01:33.465","Text":"I\u0027ll do that down here."},{"Start":"01:33.465 ","End":"01:35.640","Text":"Let\u0027s get going then."},{"Start":"01:35.640 ","End":"01:40.420","Text":"The integral of tangent squared of xdx,"},{"Start":"01:40.430 ","End":"01:45.315","Text":"and that\u0027s what\u0027s going to give us our v. Might even write that down."},{"Start":"01:45.315 ","End":"01:49.250","Text":"My v from here is equal to that."},{"Start":"01:49.250 ","End":"01:50.810","Text":"I removed the highlighting."},{"Start":"01:50.810 ","End":"01:53.465","Text":"I think it may be more confusing than helpful."},{"Start":"01:53.465 ","End":"01:55.835","Text":"What I want to do is a little bit of"},{"Start":"01:55.835 ","End":"02:00.380","Text":"trigonometric identities and simplification at the side."},{"Start":"02:00.380 ","End":"02:05.950","Text":"I want to simplify tangent squared x in a more convenient form."},{"Start":"02:05.950 ","End":"02:10.925","Text":"But first, if I use the formula that tangent is sine over cosine,"},{"Start":"02:10.925 ","End":"02:17.115","Text":"I can get sine squared x over cosine squared x."},{"Start":"02:17.115 ","End":"02:25.285","Text":"Then I can use another identity which says that sine squared is 1 minus cosine squared."},{"Start":"02:25.285 ","End":"02:28.095","Text":"Then I can break this up into 2 bits,"},{"Start":"02:28.095 ","End":"02:33.255","Text":"and say that this is 1 over cosine squared"},{"Start":"02:33.255 ","End":"02:38.870","Text":"x minus cosine squared x over cosine squared x,"},{"Start":"02:38.870 ","End":"02:40.535","Text":"which is just 1."},{"Start":"02:40.535 ","End":"02:43.455","Text":"This is much more convenient."},{"Start":"02:43.455 ","End":"02:50.060","Text":"Now, I\u0027m going to go back here and write this as the integral"},{"Start":"02:50.060 ","End":"02:58.615","Text":"of 1 over cosine squared x minus 1dx."},{"Start":"02:58.615 ","End":"03:02.720","Text":"Now, I have a minus sign here,"},{"Start":"03:02.720 ","End":"03:07.685","Text":"and I can break this up into 2 integrals with a minus between them."},{"Start":"03:07.685 ","End":"03:14.790","Text":"So this is the integral of 1 over cosine squared x dx,"},{"Start":"03:14.790 ","End":"03:18.140","Text":"that\u0027s 1 integral, minus the other integral,"},{"Start":"03:18.140 ","End":"03:20.360","Text":"which is just 1dx."},{"Start":"03:20.360 ","End":"03:26.810","Text":"Now, this is good because 1 over cosine squared is an immediate integral."},{"Start":"03:26.810 ","End":"03:29.750","Text":"The integral of this is the tangent of x."},{"Start":"03:29.750 ","End":"03:33.260","Text":"The integral of 1 is just x,"},{"Start":"03:33.260 ","End":"03:35.735","Text":"so I get tangent x minus x."},{"Start":"03:35.735 ","End":"03:40.475","Text":"This is not the time I\u0027m going to add plus c because we do that just once at the end."},{"Start":"03:40.475 ","End":"03:45.500","Text":"But at least we\u0027ve found v. Now that we\u0027ve found v,"},{"Start":"03:45.500 ","End":"03:53.645","Text":"I can put the v in here and in here and then get back to this line."},{"Start":"03:53.645 ","End":"04:00.425","Text":"What we get is tangent x minus x,"},{"Start":"04:00.425 ","End":"04:06.774","Text":"and this is also tangent x minus x."},{"Start":"04:06.774 ","End":"04:11.540","Text":"Now I\u0027m going to continue down over here from"},{"Start":"04:11.540 ","End":"04:16.250","Text":"this line and we get that our original integral"},{"Start":"04:16.250 ","End":"04:26.235","Text":"is equal to x times tangent x minus x from here, minus."},{"Start":"04:26.235 ","End":"04:28.070","Text":"Now I have the integral of a difference,"},{"Start":"04:28.070 ","End":"04:30.410","Text":"I can just split up this minus."},{"Start":"04:30.410 ","End":"04:33.740","Text":"I have minus an integral and then minus, minus is plus."},{"Start":"04:33.740 ","End":"04:44.650","Text":"I have minus the integral of tangent x and then plus the integral of x."},{"Start":"04:45.080 ","End":"04:49.560","Text":"Once again, I think I\u0027ll do a side exercise."},{"Start":"04:49.560 ","End":"04:55.550","Text":"Just like we had a side exercise for the integral of tangent squared x,"},{"Start":"04:55.550 ","End":"05:04.070","Text":"let\u0027s do another side exercise for the integral of tangent xdx."},{"Start":"05:04.070 ","End":"05:10.340","Text":"I\u0027ll highlight this. Now, this 1 is also not very difficult."},{"Start":"05:10.340 ","End":"05:14.390","Text":"I can again use trigonometry to rewrite it as the"},{"Start":"05:14.390 ","End":"05:22.830","Text":"integral of d sine x over the cosine of xdx."},{"Start":"05:23.120 ","End":"05:31.460","Text":"It reminds me very strongly of the formula that the integral of f prime over"},{"Start":"05:31.460 ","End":"05:40.970","Text":"f dx is natural log of absolute value of f. If my f were cosine x,"},{"Start":"05:40.970 ","End":"05:44.795","Text":"the f prime would have to be minus sine x."},{"Start":"05:44.795 ","End":"05:46.430","Text":"So tell you what?"},{"Start":"05:46.430 ","End":"05:52.040","Text":"Let\u0027s just put a minus here and a minus here and I think that will take care of that."},{"Start":"05:52.040 ","End":"05:57.825","Text":"Now we will get that this is equal to minus,"},{"Start":"05:57.825 ","End":"06:01.295","Text":"and then we have the formula here with f being cosine,"},{"Start":"06:01.295 ","End":"06:06.765","Text":"natural log of cosine x."},{"Start":"06:06.765 ","End":"06:09.405","Text":"That\u0027s the other side exercise."},{"Start":"06:09.405 ","End":"06:13.440","Text":"Now I\u0027m going to put this tangent x,"},{"Start":"06:13.440 ","End":"06:15.559","Text":"this integral, in here."},{"Start":"06:15.559 ","End":"06:18.020","Text":"This is what we just computed at the side."},{"Start":"06:18.020 ","End":"06:22.265","Text":"Let\u0027s get back to this line and write it below."},{"Start":"06:22.265 ","End":"06:27.870","Text":"What we get is x tangent x minus"},{"Start":"06:27.870 ","End":"06:35.540","Text":"x minus this integral we\u0027ve already computed as here."},{"Start":"06:35.540 ","End":"06:37.535","Text":"We have a minus again,"},{"Start":"06:37.535 ","End":"06:44.970","Text":"so this is going to make this a plus natural log of cosine x."},{"Start":"06:44.970 ","End":"06:47.105","Text":"Then we have this bit,"},{"Start":"06:47.105 ","End":"06:48.530","Text":"which is an immediate thing,"},{"Start":"06:48.530 ","End":"06:50.630","Text":"it\u0027s just 1/2x squared."},{"Start":"06:50.630 ","End":"06:58.370","Text":"Then plus c. I think at last we\u0027re done unless there\u0027s some simplifying to do."},{"Start":"06:58.370 ","End":"07:02.640","Text":"Now I\u0027ll just leave the answer like that. We\u0027re done."}],"ID":4436},{"Watched":false,"Name":"Exercise 24","Duration":"12m 35s","ChapterTopicVideoID":4426,"CourseChapterTopicPlaylistID":3684,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"We have to compute the following integral which I\u0027ve copied down here,"},{"Start":"00:03.930 ","End":"00:09.480","Text":"and I got to warn you this is going to be a bit lengthy because as it turns out,"},{"Start":"00:09.480 ","End":"00:13.605","Text":"we\u0027re going to have to do integration by parts, but 4 times."},{"Start":"00:13.605 ","End":"00:18.380","Text":"I\u0027ll get back to that. First, let me write the formula for integration by parts."},{"Start":"00:18.380 ","End":"00:29.430","Text":"I have the integral of udv equals uv minus the integral of vdu."},{"Start":"00:29.720 ","End":"00:34.430","Text":"Allow me also to rewrite this instead of as a square root,"},{"Start":"00:34.430 ","End":"00:36.410","Text":"as a power of a 1/2."},{"Start":"00:36.410 ","End":"00:39.035","Text":"Let me tell you in general what\u0027s going to happen."},{"Start":"00:39.035 ","End":"00:41.209","Text":"We\u0027re going to do this by parts,"},{"Start":"00:41.209 ","End":"00:43.670","Text":"but after we\u0027ve done the first round,"},{"Start":"00:43.670 ","End":"00:45.785","Text":"we\u0027re going to get something similar to this,"},{"Start":"00:45.785 ","End":"00:49.355","Text":"but with a 3 here and with a 1.5 Here."},{"Start":"00:49.355 ","End":"00:52.535","Text":"The next time we\u0027re going to get 1 lower, 2 here,"},{"Start":"00:52.535 ","End":"00:55.690","Text":"and 2.5 here, then 1 here,"},{"Start":"00:55.690 ","End":"00:58.320","Text":"and 3.5 here, and so on."},{"Start":"00:58.320 ","End":"01:02.539","Text":"We\u0027re going to keep doing it by parts and this process will eventually"},{"Start":"01:02.539 ","End":"01:06.965","Text":"terminate when we get a constant here and then we won\u0027t need it by parts."},{"Start":"01:06.965 ","End":"01:08.765","Text":"That\u0027s a general strategy."},{"Start":"01:08.765 ","End":"01:11.015","Text":"There will be 4 integrations by part,"},{"Start":"01:11.015 ","End":"01:12.500","Text":"4 because of this 4,"},{"Start":"01:12.500 ","End":"01:17.345","Text":"because it takes 4 differentiations to get rid of the exponent."},{"Start":"01:17.345 ","End":"01:19.340","Text":"Let\u0027s begin then."},{"Start":"01:19.340 ","End":"01:20.870","Text":"We need a u and a dv."},{"Start":"01:20.870 ","End":"01:23.630","Text":"I\u0027ve already hinted, this is the o ne we\u0027re going to be differentiating,"},{"Start":"01:23.630 ","End":"01:25.715","Text":"so this is going to be our u,"},{"Start":"01:25.715 ","End":"01:28.235","Text":"and therefore this is going to be dv."},{"Start":"01:28.235 ","End":"01:37.635","Text":"What we want here is uv minus the integral of v,"},{"Start":"01:37.635 ","End":"01:39.705","Text":"something here will be v,"},{"Start":"01:39.705 ","End":"01:42.960","Text":"and something here will be du."},{"Start":"01:42.960 ","End":"01:45.465","Text":"Well, let\u0027s see what that is."},{"Start":"01:45.465 ","End":"01:47.340","Text":"U, I just copy,"},{"Start":"01:47.340 ","End":"01:51.930","Text":"x plus 1 to the power of 4."},{"Start":"01:51.930 ","End":"01:58.100","Text":"V means that I have to do an integration of dv."},{"Start":"01:58.100 ","End":"02:00.065","Text":"Using the exponent rule,"},{"Start":"02:00.065 ","End":"02:02.480","Text":"noting that the inner derivative is just 1,"},{"Start":"02:02.480 ","End":"02:05.315","Text":"so I raise this power of 1 and divide by it."},{"Start":"02:05.315 ","End":"02:10.335","Text":"I get x plus 2 to the power of 1.5,"},{"Start":"02:10.335 ","End":"02:15.380","Text":"let\u0027s write that as 3 over 2 and divide it by 3 over 2."},{"Start":"02:15.380 ","End":"02:18.365","Text":"Same thing here. This is also v,"},{"Start":"02:18.365 ","End":"02:25.745","Text":"so x plus 2 to the power of 3 over 2 divided by 3 over 2."},{"Start":"02:25.745 ","End":"02:32.190","Text":"Now du, we get from u by differentiating this,"},{"Start":"02:32.190 ","End":"02:35.010","Text":"and then of course we add dx at the end."},{"Start":"02:35.010 ","End":"02:37.470","Text":"It\u0027s like something to the fourth,"},{"Start":"02:37.470 ","End":"02:42.560","Text":"4 times x plus 1 to the power of 3."},{"Start":"02:42.560 ","End":"02:44.150","Text":"There\u0027s no inner derivative,"},{"Start":"02:44.150 ","End":"02:48.410","Text":"or at least there is that it\u0027s 1 and dx."},{"Start":"02:48.410 ","End":"02:50.915","Text":"Once I\u0027ve taken care of constants,"},{"Start":"02:50.915 ","End":"02:54.350","Text":"you\u0027ll see what I mean that we get something very similar to this,"},{"Start":"02:54.350 ","End":"02:57.770","Text":"but here o ne lower and here o ne upper in the exponent."},{"Start":"02:57.770 ","End":"02:59.795","Text":"Let me just simplify this."},{"Start":"02:59.795 ","End":"03:02.330","Text":"Instead of dividing by 3 over 2,"},{"Start":"03:02.330 ","End":"03:05.525","Text":"we can of course, multiply by 2/3."},{"Start":"03:05.525 ","End":"03:10.315","Text":"So we have 2/3 of x plus 1^4."},{"Start":"03:10.315 ","End":"03:13.875","Text":"Here also the 3 over 2 comes up as 2/3,"},{"Start":"03:13.875 ","End":"03:15.030","Text":"but together with the 4,"},{"Start":"03:15.030 ","End":"03:19.680","Text":"it\u0027s 8/3 and I can bring the 8/3 out front."},{"Start":"03:19.680 ","End":"03:23.070","Text":"What I\u0027m left with is the integral of just this times this,"},{"Start":"03:23.070 ","End":"03:28.445","Text":"allow me to reverse the order because I want the x plus 1 first, just like here."},{"Start":"03:28.445 ","End":"03:33.965","Text":"So x plus 1 cubed times"},{"Start":"03:33.965 ","End":"03:39.110","Text":"x plus 2^3 over 2 dx."},{"Start":"03:39.110 ","End":"03:44.270","Text":"Now you see what I meant before that we\u0027ve got an integration by parts here."},{"Start":"03:44.270 ","End":"03:46.540","Text":"We\u0027ll get another integration by parts here,"},{"Start":"03:46.540 ","End":"03:49.280","Text":"but each time this exponent will get lower,"},{"Start":"03:49.280 ","End":"03:50.645","Text":"we just lowered it to 3,"},{"Start":"03:50.645 ","End":"03:53.530","Text":"next round we\u0027ll lower it to 2, and so on."},{"Start":"03:53.530 ","End":"03:58.445","Text":"What I\u0027m going to do is do this o ne separately at the side,"},{"Start":"03:58.445 ","End":"04:02.990","Text":"so I\u0027ll highlight that this o ne I\u0027m going"},{"Start":"04:02.990 ","End":"04:08.050","Text":"to do separately and then come back and substitute it."},{"Start":"04:08.050 ","End":"04:13.910","Text":"What we\u0027ll get is the integral of x plus"},{"Start":"04:13.910 ","End":"04:23.805","Text":"1 cubed times x plus 2^3 over 2 dx is equal to."},{"Start":"04:23.805 ","End":"04:26.750","Text":"Now, I\u0027m going to do this a bit quicker the second time,"},{"Start":"04:26.750 ","End":"04:28.820","Text":"I won\u0027t bother with writing all du and dv,"},{"Start":"04:28.820 ","End":"04:30.080","Text":"it\u0027s going to be the same as here."},{"Start":"04:30.080 ","End":"04:32.750","Text":"This is u and this is dv."},{"Start":"04:32.750 ","End":"04:36.375","Text":"So uv is what I need,"},{"Start":"04:36.375 ","End":"04:38.340","Text":"u will be this."},{"Start":"04:38.340 ","End":"04:40.290","Text":"I\u0027ll just write du and the dv,"},{"Start":"04:40.290 ","End":"04:42.990","Text":"perhaps, that\u0027s all that I need to do."},{"Start":"04:42.990 ","End":"04:47.880","Text":"I need uv, x plus 1 cubed,"},{"Start":"04:47.880 ","End":"04:50.190","Text":"and v, just like before,"},{"Start":"04:50.190 ","End":"04:53.205","Text":"instead of 1/2, we have 3 over 2,"},{"Start":"04:53.205 ","End":"04:58.925","Text":"so we have this time to the power of 5 over 2 if I raise this by 1."},{"Start":"04:58.925 ","End":"05:04.100","Text":"I get x plus 2 to the power of 5 over"},{"Start":"05:04.100 ","End":"05:10.430","Text":"2 over 5 over 2 minus,"},{"Start":"05:10.430 ","End":"05:12.890","Text":"and again, the same thing,"},{"Start":"05:12.890 ","End":"05:15.870","Text":"integral of vdu, so it\u0027s the same thing."},{"Start":"05:15.870 ","End":"05:23.555","Text":"It\u0027s x plus 2 to the 5 over 2 over 5 over 2."},{"Start":"05:23.555 ","End":"05:28.745","Text":"Then du is just the derivative of,"},{"Start":"05:28.745 ","End":"05:36.180","Text":"this bit is going to be 3x plus 1 squared."},{"Start":"05:36.180 ","End":"05:40.235","Text":"You see each time the power of x plus 1 is decreasing."},{"Start":"05:40.235 ","End":"05:42.620","Text":"Simplifying this a little bit,"},{"Start":"05:42.620 ","End":"05:49.280","Text":"I get 2/5 x plus 1 cubed,"},{"Start":"05:49.280 ","End":"05:54.419","Text":"x plus 2^5 over 2."},{"Start":"05:54.419 ","End":"06:00.675","Text":"Now, here again, 2 over 5 times 3 is 6 over 5,"},{"Start":"06:00.675 ","End":"06:03.525","Text":"6 over 5, the integral,"},{"Start":"06:03.525 ","End":"06:06.210","Text":"and I\u0027ll reverse the order of the factors,"},{"Start":"06:06.210 ","End":"06:12.640","Text":"so it\u0027s x plus 1 squared times x"},{"Start":"06:12.640 ","End":"06:19.865","Text":"plus 2 to the power of 5 over 2 dx."},{"Start":"06:19.865 ","End":"06:23.304","Text":"This o ne, I\u0027ll use a different color."},{"Start":"06:23.304 ","End":"06:26.560","Text":"I\u0027m also going to do separately."},{"Start":"06:26.560 ","End":"06:28.600","Text":"This will be the next bit,"},{"Start":"06:28.600 ","End":"06:31.945","Text":"and I\u0027ll do that over here and so on."},{"Start":"06:31.945 ","End":"06:34.855","Text":"There may be 4 of these."},{"Start":"06:34.855 ","End":"06:40.390","Text":"I\u0027ll leave it to you to disentangle it at the end, so let\u0027s continue."},{"Start":"06:40.390 ","End":"06:44.140","Text":"What I want is to do this bit here."},{"Start":"06:44.140 ","End":"06:52.190","Text":"The integral of x plus 1 squared,"},{"Start":"06:52.190 ","End":"06:56.915","Text":"x plus 2^5 over 2 dx."},{"Start":"06:56.915 ","End":"06:59.390","Text":"Again, it\u0027s going to be by parts."},{"Start":"06:59.390 ","End":"07:03.605","Text":"This is going to be u, this is going to be dv."},{"Start":"07:03.605 ","End":"07:06.965","Text":"I\u0027m just saying now at the end when we get what this is,"},{"Start":"07:06.965 ","End":"07:09.395","Text":"we substitute it into here."},{"Start":"07:09.395 ","End":"07:11.630","Text":"Once we got this, we can compute this and"},{"Start":"07:11.630 ","End":"07:14.840","Text":"substitute it into here and then so on and so on."},{"Start":"07:14.840 ","End":"07:18.365","Text":"We\u0027re actually going to be working from down upwards."},{"Start":"07:18.365 ","End":"07:19.845","Text":"When we don\u0027t have an integral,"},{"Start":"07:19.845 ","End":"07:22.625","Text":"we\u0027ll keep substituting backwards, and so on."},{"Start":"07:22.625 ","End":"07:24.980","Text":"I\u0027m telling you this to prepare you that at the end"},{"Start":"07:24.980 ","End":"07:27.215","Text":"I\u0027m not going to do all this hard work,"},{"Start":"07:27.215 ","End":"07:29.210","Text":"which is completely routine."},{"Start":"07:29.210 ","End":"07:33.770","Text":"I\u0027m going to get to the end and I\u0027ll leave you at the end to do the substitutions."},{"Start":"07:33.770 ","End":"07:39.170","Text":"Continuing, just as before, it\u0027s fairly routine."},{"Start":"07:39.170 ","End":"07:41.330","Text":"Now I write here u,"},{"Start":"07:41.330 ","End":"07:45.830","Text":"which is x plus 1 squared times v. Again,"},{"Start":"07:45.830 ","End":"07:50.585","Text":"I\u0027m going to raise the exponent by 1 and get x plus 2,"},{"Start":"07:50.585 ","End":"07:56.585","Text":"instead of 5 over 2 it\u0027s going to be 7 over 2 and divide it by 7 over 2."},{"Start":"07:56.585 ","End":"07:58.865","Text":"Then the integral again, v,"},{"Start":"07:58.865 ","End":"08:05.540","Text":"x plus 2^7 over 2 over 7 over 2 times du,"},{"Start":"08:05.540 ","End":"08:07.130","Text":"which is just the derivative of this,"},{"Start":"08:07.130 ","End":"08:13.950","Text":"which is twice x plus 1 dx."},{"Start":"08:13.950 ","End":"08:18.740","Text":"A little simplification just with the numbers 7 over 2,"},{"Start":"08:18.740 ","End":"08:25.230","Text":"so it comes to the front as 2/7 of x plus 1 squared x"},{"Start":"08:25.230 ","End":"08:32.240","Text":"plus 2^7 over 2 minus the integral."},{"Start":"08:32.240 ","End":"08:34.760","Text":"Now this 7 over 2 is 2/7,"},{"Start":"08:34.760 ","End":"08:37.310","Text":"with this 2 makes it 4/7."},{"Start":"08:37.310 ","End":"08:40.340","Text":"So 4/7, the integral of this times this,"},{"Start":"08:40.340 ","End":"08:42.290","Text":"and as before, I\u0027ll reverse the order."},{"Start":"08:42.290 ","End":"08:43.760","Text":"It\u0027s x plus 1,"},{"Start":"08:43.760 ","End":"08:45.920","Text":"this time only to the power of 1,"},{"Start":"08:45.920 ","End":"08:51.830","Text":"x plus 2 to the power of 7 over 2 dx. You see the progression."},{"Start":"08:51.830 ","End":"08:54.870","Text":"We started off here with 4 and this is a 1/2,"},{"Start":"08:54.870 ","End":"08:58.830","Text":"and then we got 3 and 1.5,"},{"Start":"08:58.830 ","End":"09:04.285","Text":"2 and 2.5, and this time we\u0027re going to get 1 and 3.5."},{"Start":"09:04.285 ","End":"09:08.150","Text":"Choose a different color for the highlighting."},{"Start":"09:08.150 ","End":"09:10.015","Text":"Let\u0027s say this color,"},{"Start":"09:10.015 ","End":"09:12.419","Text":"and I\u0027m going to do this o ne at the side,"},{"Start":"09:12.419 ","End":"09:17.450","Text":"just rewrite that as integral of x plus"},{"Start":"09:17.450 ","End":"09:25.740","Text":"1 times x plus 2^7 over 2 dx."},{"Start":"09:25.740 ","End":"09:28.520","Text":"Yeah. Once again, I\u0027ll just like to look here."},{"Start":"09:28.520 ","End":"09:31.355","Text":"We had 4, 3, 2, 1."},{"Start":"09:31.355 ","End":"09:36.995","Text":"Next round we\u0027ll have blastoff and we\u0027ll stop the integration by parts."},{"Start":"09:36.995 ","End":"09:38.735","Text":"Meanwhile, this is equal to,"},{"Start":"09:38.735 ","End":"09:41.215","Text":"so I\u0027m going to let this be u,"},{"Start":"09:41.215 ","End":"09:44.665","Text":"this is a fourth integration by parts,"},{"Start":"09:44.665 ","End":"09:48.215","Text":"and this will be my dv,"},{"Start":"09:48.215 ","End":"09:54.525","Text":"u, which is x plus 1 times v,"},{"Start":"09:54.525 ","End":"10:01.230","Text":"which is raise the power by 1,"},{"Start":"10:01.230 ","End":"10:09.350","Text":"and I\u0027ll get x plus 2^9 over 2 over 9 over 2."},{"Start":"10:09.350 ","End":"10:14.060","Text":"That\u0027s my v minus the integral of v. Same thing as here,"},{"Start":"10:14.060 ","End":"10:22.655","Text":"x plus 2^9 over 2 over 9 over 2 times du."},{"Start":"10:22.655 ","End":"10:24.140","Text":"If u is x plus 1,"},{"Start":"10:24.140 ","End":"10:26.930","Text":"du is just dx."},{"Start":"10:26.930 ","End":"10:34.300","Text":"Rewriting this, we get 9 over 2 comes to the front as 2/9."},{"Start":"10:34.300 ","End":"10:39.145","Text":"So 2/9 x plus 1,"},{"Start":"10:39.145 ","End":"10:45.555","Text":"x plus 2^9 over 2 minus,"},{"Start":"10:45.555 ","End":"10:50.610","Text":"9 over 2 comes out in front as 2/9."},{"Start":"10:50.610 ","End":"10:59.220","Text":"The integral of x plus 2^9 over 2 dx."},{"Start":"10:59.220 ","End":"11:05.090","Text":"Here, finally, we can actually do this not by parts,"},{"Start":"11:05.090 ","End":"11:06.590","Text":"so I\u0027m just continuing,"},{"Start":"11:06.590 ","End":"11:09.485","Text":"don\u0027t need to bring it to the side or anything."},{"Start":"11:09.485 ","End":"11:14.090","Text":"This is equal to 2/9,"},{"Start":"11:14.090 ","End":"11:20.405","Text":"x plus 1, x plus 2^9 over 2,"},{"Start":"11:20.405 ","End":"11:24.380","Text":"minus 2/9, and the integral of this is just,"},{"Start":"11:24.380 ","End":"11:28.865","Text":"I raise it by 1, will be 11 over 2,"},{"Start":"11:28.865 ","End":"11:31.460","Text":"and then divide it by 11 over 2."},{"Start":"11:31.460 ","End":"11:32.810","Text":"Already you can see the pattern."},{"Start":"11:32.810 ","End":"11:34.880","Text":"Instead of dividing by 11 over 2,"},{"Start":"11:34.880 ","End":"11:37.340","Text":"I\u0027ll put 2 over 11 here,"},{"Start":"11:37.340 ","End":"11:40.650","Text":"and I\u0027ll get x plus 2^11 over 2."},{"Start":"11:40.650 ","End":"11:41.990","Text":"Yeah, you see what I mean."},{"Start":"11:41.990 ","End":"11:44.180","Text":"Instead of 11 over 2 in the denominator,"},{"Start":"11:44.180 ","End":"11:46.310","Text":"2 over 11 in the numerator."},{"Start":"11:46.310 ","End":"11:50.795","Text":"I\u0027m not putting the plus c because we do that once at the end."},{"Start":"11:50.795 ","End":"11:52.895","Text":"I\u0027m basically done."},{"Start":"11:52.895 ","End":"11:58.325","Text":"What you need to do is to take this,"},{"Start":"11:58.325 ","End":"12:02.809","Text":"which is actually the expansion of the purple,"},{"Start":"12:02.809 ","End":"12:07.550","Text":"or perhaps I could keep coloring it."},{"Start":"12:07.550 ","End":"12:09.515","Text":"This is this, which is this,"},{"Start":"12:09.515 ","End":"12:11.014","Text":"but we work backwards."},{"Start":"12:11.014 ","End":"12:13.610","Text":"What we do is we now have this expression,"},{"Start":"12:13.610 ","End":"12:16.835","Text":"so we\u0027ve computed the magenta expression."},{"Start":"12:16.835 ","End":"12:20.055","Text":"You write this instead of the magenta."},{"Start":"12:20.055 ","End":"12:21.650","Text":"When you\u0027re done with that,"},{"Start":"12:21.650 ","End":"12:26.645","Text":"we will have had this computed and we substitute it into the yellow."},{"Start":"12:26.645 ","End":"12:28.715","Text":"When we\u0027ve simplified this,"},{"Start":"12:28.715 ","End":"12:32.420","Text":"we can then substitute it into the cyan or whatever."},{"Start":"12:32.420 ","End":"12:34.365","Text":"I consider myself finished."},{"Start":"12:34.365 ","End":"12:36.520","Text":"We are done."}],"ID":4437}],"Thumbnail":null,"ID":3684}]

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