Introduction to Integration by Substitution
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[{"Name":"Introduction to Integration by Substitution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Integration by Substitution","Duration":"17m 5s","ChapterTopicVideoID":3108,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3108.jpeg","UploadDate":"2019-12-11T21:02:28.9500000","DurationForVideoObject":"PT17M5S","Description":null,"MetaTitle":"Substitution: Video + Workbook | Proprep","MetaDescription":"Integration by Substitution - Introduction to Integration by Substitution. 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-substitution/introduction-to-integration-by-substitution/vid3120","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.330","Text":"In this clip, we\u0027re going to learn a very useful technique for doing integrals,"},{"Start":"00:05.330 ","End":"00:08.250","Text":"and it\u0027s called integration by substitution."},{"Start":"00:08.250 ","End":"00:10.620","Text":"We\u0027ll start right away with an example."},{"Start":"00:10.620 ","End":"00:16.890","Text":"I have to find the indefinite integral of x cubed"},{"Start":"00:16.890 ","End":"00:24.615","Text":"plus 1^10 times x squared dx."},{"Start":"00:24.615 ","End":"00:30.270","Text":"Now, nothing we\u0027ve learned up till now is going to help us to solve this."},{"Start":"00:30.270 ","End":"00:32.865","Text":"There are no product rules, no exponent."},{"Start":"00:32.865 ","End":"00:35.855","Text":"I mean, there\u0027s just nothing available."},{"Start":"00:35.855 ","End":"00:37.880","Text":"The only thing you could possibly do would be to"},{"Start":"00:37.880 ","End":"00:40.085","Text":"expand the whole thing to the power of 10,"},{"Start":"00:40.085 ","End":"00:42.065","Text":"and that\u0027s not feasible."},{"Start":"00:42.065 ","End":"00:43.840","Text":"We\u0027re going to learn a new technique,"},{"Start":"00:43.840 ","End":"00:45.950","Text":"and it\u0027s called substitution."},{"Start":"00:45.950 ","End":"00:50.730","Text":"The first step is to take part of our integral."},{"Start":"00:50.730 ","End":"00:53.660","Text":"This is where the art of learning what to choose."},{"Start":"00:53.660 ","End":"01:00.155","Text":"But in this case, a good choice would be x cubed plus 1 and let it be another variable,"},{"Start":"01:00.155 ","End":"01:05.180","Text":"let\u0027s say t. I\u0027m letting x cubed plus 1"},{"Start":"01:05.180 ","End":"01:10.370","Text":"equal a new variable t. This is the actual substitution."},{"Start":"01:10.370 ","End":"01:15.060","Text":"This part, I\u0027m going to call it substitute."},{"Start":"01:15.060 ","End":"01:20.660","Text":"The same steps will appear in all the substitution exercises."},{"Start":"01:20.660 ","End":"01:26.055","Text":"Now if I just substitute x cubed plus 1 is t,"},{"Start":"01:26.055 ","End":"01:31.215","Text":"what I\u0027ll get will be the integral"},{"Start":"01:31.215 ","End":"01:37.155","Text":"of t_10 times x squared dx."},{"Start":"01:37.155 ","End":"01:38.820","Text":"This is not very good."},{"Start":"01:38.820 ","End":"01:40.895","Text":"There\u0027s 2 major problems."},{"Start":"01:40.895 ","End":"01:44.674","Text":"First of all, I\u0027ve got a mixture of t and x,"},{"Start":"01:44.674 ","End":"01:46.895","Text":"and that\u0027s not good."},{"Start":"01:46.895 ","End":"01:49.324","Text":"I need all t. Also,"},{"Start":"01:49.324 ","End":"01:50.765","Text":"I have here a dx."},{"Start":"01:50.765 ","End":"01:52.310","Text":"When I\u0027m doing an integral with t,"},{"Start":"01:52.310 ","End":"01:56.470","Text":"I need the dt, I need to manufacturer dt out of something."},{"Start":"01:56.470 ","End":"02:00.320","Text":"Next thing we do is to differentiate."},{"Start":"02:00.320 ","End":"02:02.630","Text":"When we differentiate this,"},{"Start":"02:02.630 ","End":"02:05.700","Text":"we get 3x squared."},{"Start":"02:05.740 ","End":"02:11.165","Text":"Here, I get 1."},{"Start":"02:11.165 ","End":"02:13.400","Text":"But here\u0027s the thing we do,"},{"Start":"02:13.400 ","End":"02:15.530","Text":"it\u0027s just technical at the moment."},{"Start":"02:15.530 ","End":"02:17.345","Text":"On the side of the x,"},{"Start":"02:17.345 ","End":"02:20.290","Text":"we artificially add dx,"},{"Start":"02:20.290 ","End":"02:21.930","Text":"and on the side of the t,"},{"Start":"02:21.930 ","End":"02:24.040","Text":"we artificially add dt."},{"Start":"02:24.040 ","End":"02:26.645","Text":"There\u0027s very good basis for doing this."},{"Start":"02:26.645 ","End":"02:28.595","Text":"It\u0027s justified. But at the moment,"},{"Start":"02:28.595 ","End":"02:30.860","Text":"for you, it\u0027s just a technique."},{"Start":"02:30.860 ","End":"02:37.350","Text":"The third thing to do is to extract or isolate dx."},{"Start":"02:40.930 ","End":"02:44.940","Text":"What that means is just get dx on its own."},{"Start":"02:44.940 ","End":"02:54.490","Text":"Simple algebra, dx is equal to dt over 3x squared."},{"Start":"02:57.590 ","End":"03:03.010","Text":"Now what we do is to start actually substituting."},{"Start":"03:03.010 ","End":"03:05.140","Text":"There\u0027s 2 things we substitute."},{"Start":"03:05.140 ","End":"03:07.270","Text":"Wherever we see x cubed plus 1,"},{"Start":"03:07.270 ","End":"03:11.010","Text":"we put t, and wherever we see dx, we put this."},{"Start":"03:11.010 ","End":"03:14.265","Text":"In our case, here\u0027s the x cubed plus 1,"},{"Start":"03:14.265 ","End":"03:16.725","Text":"and here\u0027s the dx."},{"Start":"03:16.725 ","End":"03:22.275","Text":"What we get now is the integral."},{"Start":"03:22.275 ","End":"03:26.355","Text":"X cubed plus 1 is t^10."},{"Start":"03:26.355 ","End":"03:28.830","Text":"X squared stays as it is."},{"Start":"03:28.830 ","End":"03:36.780","Text":"The dx we substitute from here is dt/3x squared."},{"Start":"03:36.780 ","End":"03:41.955","Text":"Now look, dx squared and dx squared cancel,"},{"Start":"03:41.955 ","End":"03:44.455","Text":"so what I\u0027m left with is 1/3,"},{"Start":"03:44.455 ","End":"03:50.660","Text":"which I can take in front of the integral of t^10 dt."},{"Start":"03:50.660 ","End":"03:52.925","Text":"This is something we know how to do."},{"Start":"03:52.925 ","End":"03:58.325","Text":"This is equal, the integral of this is t^11/11."},{"Start":"03:58.325 ","End":"04:02.180","Text":"Altogether I get 1/11."},{"Start":"04:02.180 ","End":"04:04.205","Text":"11 goes with the 3,"},{"Start":"04:04.205 ","End":"04:08.480","Text":"which is 33, t^11,"},{"Start":"04:08.480 ","End":"04:11.990","Text":"and finally plus c. However,"},{"Start":"04:11.990 ","End":"04:17.060","Text":"this is not the end of the affair because we have now an answer in t,"},{"Start":"04:17.060 ","End":"04:19.580","Text":"but the original exercise was in x."},{"Start":"04:19.580 ","End":"04:24.110","Text":"What we have to do finally is another substitution to substitute back."},{"Start":"04:24.110 ","End":"04:25.400","Text":"I\u0027m not going to give it a label,"},{"Start":"04:25.400 ","End":"04:27.350","Text":"but just remember to do this,"},{"Start":"04:27.350 ","End":"04:35.030","Text":"not to leave the answer in the form of t. This is now equal to 1/33."},{"Start":"04:35.030 ","End":"04:38.460","Text":"T is x cubed plus 1^11,"},{"Start":"04:41.200 ","End":"04:46.020","Text":"plus c. This is the answer."},{"Start":"04:46.030 ","End":"04:48.620","Text":"Let\u0027s do another example."},{"Start":"04:48.620 ","End":"04:51.260","Text":"This time I\u0027ll take the integral of"},{"Start":"04:51.260 ","End":"05:00.930","Text":"the natural log of x_4/x dx."},{"Start":"05:00.930 ","End":"05:03.345","Text":"We need to substitute something."},{"Start":"05:03.345 ","End":"05:09.260","Text":"I think what is most natural to do is to let natural log of"},{"Start":"05:09.260 ","End":"05:17.670","Text":"x equals t. Next thing we have to do is differentiate."},{"Start":"05:17.670 ","End":"05:23.090","Text":"The derivative of natural log of x is 1/x."},{"Start":"05:23.090 ","End":"05:26.150","Text":"Remember, we artificially add a dx,"},{"Start":"05:26.150 ","End":"05:30.995","Text":"and the derivative of t with respect to t is 1,"},{"Start":"05:30.995 ","End":"05:34.320","Text":"and we add a dt here."},{"Start":"05:34.320 ","End":"05:38.880","Text":"Now there are 2 quantities that we want to substitute,"},{"Start":"05:38.880 ","End":"05:41.880","Text":"1 of them is natural log of x,"},{"Start":"05:41.880 ","End":"05:45.210","Text":"and the other 1 is the dx."},{"Start":"05:45.210 ","End":"05:48.535","Text":"I forgot the extract dx."},{"Start":"05:48.535 ","End":"05:53.070","Text":"Dx is equal to xdt."},{"Start":"05:53.690 ","End":"05:58.110","Text":"This is what I wanted to highlight."},{"Start":"05:58.110 ","End":"06:08.110","Text":"These 2 things, we substitute in the function and we get a new integral, t^4."},{"Start":"06:08.110 ","End":"06:10.875","Text":"The x just stays as is."},{"Start":"06:10.875 ","End":"06:14.920","Text":"The dx gets substituted to xdt."},{"Start":"06:17.480 ","End":"06:21.330","Text":"Again, lucky, the x cancels."},{"Start":"06:21.330 ","End":"06:24.970","Text":"I have just the integral of t^4 dt."},{"Start":"06:25.430 ","End":"06:35.170","Text":"This equals t^5/5 plus c. Remember,"},{"Start":"06:35.170 ","End":"06:38.090","Text":"our exercise was an integral in x,"},{"Start":"06:38.090 ","End":"06:40.460","Text":"and here we have t. At the end,"},{"Start":"06:40.460 ","End":"06:42.710","Text":"we substitute back, instead of t,"},{"Start":"06:42.710 ","End":"06:44.840","Text":"we put the natural log of x."},{"Start":"06:44.840 ","End":"06:55.580","Text":"The answer is natural log of x^5 over 5. That\u0027s it."},{"Start":"06:55.580 ","End":"06:59.660","Text":"I\u0027m going to do some more examples that contain"},{"Start":"06:59.660 ","End":"07:04.010","Text":"square roots and radicals because these are very popular in exams."},{"Start":"07:04.010 ","End":"07:09.230","Text":"The first one I\u0027ll do is the integral of"},{"Start":"07:09.230 ","End":"07:18.315","Text":"x over the square root of x squared plus 1, the dx."},{"Start":"07:18.315 ","End":"07:20.640","Text":"Substitution."},{"Start":"07:20.640 ","End":"07:23.100","Text":"Let\u0027s see what we\u0027re going to substitute."},{"Start":"07:23.100 ","End":"07:28.460","Text":"I\u0027ll just write substitute and I\u0027ll have a time to think what I want to substitute."},{"Start":"07:28.460 ","End":"07:34.950","Text":"It seems to me that I should substitute the square root of x squared plus 1."},{"Start":"07:36.280 ","End":"07:42.655","Text":"That\u0027s typically what we do is the root or radical is we substituted."},{"Start":"07:42.655 ","End":"07:50.225","Text":"That will let equal to t. Then we differentiate or the derive it shorter."},{"Start":"07:50.225 ","End":"07:52.775","Text":"But I\u0027m not going to derive it straight away."},{"Start":"07:52.775 ","End":"07:54.785","Text":"I\u0027m going to square it first."},{"Start":"07:54.785 ","End":"08:01.080","Text":"I\u0027m going to say that x squared plus 1 is equal to t squared."},{"Start":"08:02.170 ","End":"08:05.465","Text":"Then if I derive,"},{"Start":"08:05.465 ","End":"08:06.980","Text":"here I get 2x,"},{"Start":"08:06.980 ","End":"08:08.720","Text":"here I get 2t."},{"Start":"08:08.720 ","End":"08:15.880","Text":"But remember, we have to add the dx on this side and the dt on this side."},{"Start":"08:15.880 ","End":"08:24.075","Text":"Then finally, I isolate or extract the dx is the next thing."},{"Start":"08:24.075 ","End":"08:27.120","Text":"We get that dx is equal,"},{"Start":"08:27.120 ","End":"08:34.715","Text":"bring this to the other side, I get tdt/x."},{"Start":"08:34.715 ","End":"08:40.255","Text":"Now, wherever we see square root of x squared plus 1,"},{"Start":"08:40.255 ","End":"08:43.345","Text":"we substitute it, and also the dx."},{"Start":"08:43.345 ","End":"08:46.840","Text":"These are the 2 things, and only these 2 things do we substitute."},{"Start":"08:46.840 ","End":"08:50.045","Text":"In this case, what I get is,"},{"Start":"08:50.045 ","End":"08:57.315","Text":"this is equal to the integral x stays x."},{"Start":"08:57.315 ","End":"09:01.200","Text":"This can be substituted to be t,"},{"Start":"09:01.200 ","End":"09:03.760","Text":"and the dx becomes"},{"Start":"09:03.760 ","End":"09:13.480","Text":"tdt/x."},{"Start":"09:13.620 ","End":"09:19.870","Text":"The x cancels fortunately and in fact even the t cancels."},{"Start":"09:19.870 ","End":"09:23.620","Text":"All I\u0027m left with is the integral of dt."},{"Start":"09:23.620 ","End":"09:26.150","Text":"Well, let\u0027s call it 1dt."},{"Start":"09:27.060 ","End":"09:29.290","Text":"This integral is easy."},{"Start":"09:29.290 ","End":"09:34.120","Text":"This is just t itself plus a constant and finally,"},{"Start":"09:34.120 ","End":"09:36.310","Text":"we go from t back to x,"},{"Start":"09:36.310 ","End":"09:40.600","Text":"and t is the square root of x squared"},{"Start":"09:40.600 ","End":"09:47.630","Text":"plus 1 plus c. Let\u0027s go with another example."},{"Start":"09:48.240 ","End":"09:51.565","Text":"This time we\u0027ll take a cube root,"},{"Start":"09:51.565 ","End":"09:59.690","Text":"the cube root of x squared plus 1 times xdx."},{"Start":"10:00.780 ","End":"10:04.480","Text":"The first step is to substitute something."},{"Start":"10:04.480 ","End":"10:10.390","Text":"What we\u0027ll do is substitute the whole cube root and that the cube root of x squared"},{"Start":"10:10.390 ","End":"10:17.155","Text":"plus 1 equal t and then we differentiate."},{"Start":"10:17.155 ","End":"10:20.410","Text":"But before we actually differentiate,"},{"Start":"10:20.410 ","End":"10:23.815","Text":"we also raise to the power of 3, makes life easier."},{"Start":"10:23.815 ","End":"10:25.825","Text":"Don\u0027t want to get rid of that cube root,"},{"Start":"10:25.825 ","End":"10:31.240","Text":"so we get that x squared plus 1 equals t cubed."},{"Start":"10:31.240 ","End":"10:37.420","Text":"Then the actual differentiation, we get that 2x,"},{"Start":"10:37.420 ","End":"10:39.775","Text":"and don\u0027t forget to add the dx,"},{"Start":"10:39.775 ","End":"10:44.785","Text":"equals 3t squared and we add the dt."},{"Start":"10:44.785 ","End":"10:49.825","Text":"Then we extract the dx,"},{"Start":"10:49.825 ","End":"10:54.640","Text":"so we\u0027re left with dx is equal to"},{"Start":"10:54.640 ","End":"11:03.380","Text":"3t squared dt over 2x."},{"Start":"11:04.080 ","End":"11:13.285","Text":"Next thing is to just mark the original substitution and the dx."},{"Start":"11:13.285 ","End":"11:15.820","Text":"These things when we find them in the original,"},{"Start":"11:15.820 ","End":"11:20.620","Text":"we want to replace them and the rest we leave alone."},{"Start":"11:20.620 ","End":"11:29.850","Text":"This is going to equal the integral of the cube root of x squared plus 1 is just t,"},{"Start":"11:29.850 ","End":"11:32.685","Text":"x I leave as it is,"},{"Start":"11:32.685 ","End":"11:41.360","Text":"and dx I replace by 3t squared dt over 2x."},{"Start":"11:41.360 ","End":"11:43.555","Text":"Let\u0027s see what we get."},{"Start":"11:43.555 ","End":"11:49.120","Text":"The x cancels fortunately and what we\u0027re left with is 3 over 2,"},{"Start":"11:49.120 ","End":"11:58.060","Text":"I\u0027ll put in front of the integral of t and t squared is t cubed and dt."},{"Start":"11:58.830 ","End":"12:01.780","Text":"Now this we know how to do."},{"Start":"12:01.780 ","End":"12:03.520","Text":"We raise the power by 1,"},{"Start":"12:03.520 ","End":"12:07.450","Text":"so it\u0027s t to the 4th and also divide by 4."},{"Start":"12:07.450 ","End":"12:09.340","Text":"Dividing by 4 will give us 3,"},{"Start":"12:09.340 ","End":"12:15.490","Text":"8s and then we\u0027ll get t to the 4th plus a constant."},{"Start":"12:15.490 ","End":"12:23.260","Text":"Then finally, we want to replace t by what it was in terms of x,"},{"Start":"12:23.260 ","End":"12:26.980","Text":"so it\u0027s the cube root of x and to the power of 4,"},{"Start":"12:26.980 ","End":"12:29.905","Text":"so I\u0027ll use the fractional exponent."},{"Start":"12:29.905 ","End":"12:33.595","Text":"This is x squared plus 1 to the power of 1/3,"},{"Start":"12:33.595 ","End":"12:37.780","Text":"so altogether I get x squared plus 1 not to the power of 1/3,"},{"Start":"12:37.780 ","End":"12:41.454","Text":"but to the power of 4/3s plus a constant,"},{"Start":"12:41.454 ","End":"12:44.695","Text":"and this is our answer."},{"Start":"12:44.695 ","End":"12:48.580","Text":"Notice how in all our examples so far we"},{"Start":"12:48.580 ","End":"12:52.120","Text":"got to a stage where there was no x\u0027s, everything canceled."},{"Start":"12:52.120 ","End":"12:56.815","Text":"It\u0027s not always that easy as the following example will show."},{"Start":"12:56.815 ","End":"13:02.830","Text":"This time the example\u0027s going to be the integral of"},{"Start":"13:02.830 ","End":"13:08.470","Text":"x cubed over square root"},{"Start":"13:08.470 ","End":"13:14.710","Text":"of x squared plus 1 dx."},{"Start":"13:14.710 ","End":"13:18.350","Text":"Now I want to substitute something."},{"Start":"13:19.290 ","End":"13:23.080","Text":"What I\u0027m going to substitute is the square root,"},{"Start":"13:23.080 ","End":"13:30.250","Text":"so we\u0027re going to let the square root of x squared plus 1 equal t,"},{"Start":"13:30.250 ","End":"13:34.570","Text":"and then we derive, differentiate."},{"Start":"13:34.570 ","End":"13:40.375","Text":"But before that, we get rid of the roots, the radicals."},{"Start":"13:40.375 ","End":"13:45.895","Text":"We get x squared plus 1 is equal to t squared."},{"Start":"13:45.895 ","End":"13:48.340","Text":"Then from here 2x,"},{"Start":"13:48.340 ","End":"13:54.520","Text":"from here 2t, but don\u0027t forget the dx and the dt."},{"Start":"13:54.520 ","End":"14:00.550","Text":"Then we extract or isolate just the dx,"},{"Start":"14:00.550 ","End":"14:03.760","Text":"so just easy, divide both sides by 2x."},{"Start":"14:03.760 ","End":"14:11.330","Text":"We get that dx is equal to tdt over x."},{"Start":"14:11.700 ","End":"14:14.755","Text":"The 2 cancel here."},{"Start":"14:14.755 ","End":"14:19.209","Text":"Now I\u0027m going to highlight the things I want to substitute."},{"Start":"14:19.209 ","End":"14:21.760","Text":"The dx has to be substituted and"},{"Start":"14:21.760 ","End":"14:28.315","Text":"the original square root of x squared plus 1 in here and everything else stays."},{"Start":"14:28.315 ","End":"14:32.975","Text":"What we get is the integral."},{"Start":"14:32.975 ","End":"14:37.190","Text":"X cubed stays x cubed."},{"Start":"14:37.190 ","End":"14:45.410","Text":"Square root of x squared plus 1 is t and dx is tdt over x."},{"Start":"14:51.150 ","End":"14:54.115","Text":"Now in this case,"},{"Start":"14:54.115 ","End":"14:57.460","Text":"the t actually cancels and in fact,"},{"Start":"14:57.460 ","End":"15:01.870","Text":"1 of the x\u0027s cancels to make this x squared."},{"Start":"15:01.870 ","End":"15:09.560","Text":"What we\u0027re left with is the integral of x squared dt."},{"Start":"15:10.260 ","End":"15:14.710","Text":"Now this seems to be a problem because we wanted an integral in"},{"Start":"15:14.710 ","End":"15:19.450","Text":"t. We have to use a bit of algebra if we can,"},{"Start":"15:19.450 ","End":"15:22.100","Text":"to try and get rid of the x."},{"Start":"15:22.950 ","End":"15:26.740","Text":"If we can\u0027t, it may be 1 or several reasons."},{"Start":"15:26.740 ","End":"15:28.975","Text":"It could be that we made an error somewhere,"},{"Start":"15:28.975 ","End":"15:32.320","Text":"or it could just be that we went creative enough,"},{"Start":"15:32.320 ","End":"15:36.715","Text":"or it could be that this integral is not amenable to substitution."},{"Start":"15:36.715 ","End":"15:39.100","Text":"Substitution can\u0027t work on all integrals."},{"Start":"15:39.100 ","End":"15:42.190","Text":"But let\u0027s try and see if we can do something algebraic."},{"Start":"15:42.190 ","End":"15:43.960","Text":"We\u0027ll look at this line here."},{"Start":"15:43.960 ","End":"15:46.644","Text":"X squared plus 1 is t squared."},{"Start":"15:46.644 ","End":"15:49.269","Text":"If x squared plus 1 is t squared,"},{"Start":"15:49.269 ","End":"15:52.555","Text":"then it\u0027s easy enough to see what x squared is."},{"Start":"15:52.555 ","End":"15:57.050","Text":"X squared will be t squared minus 1."},{"Start":"15:57.060 ","End":"16:01.675","Text":"Now we have the integral of t squared minus 1dt and all is well,"},{"Start":"16:01.675 ","End":"16:04.480","Text":"so that was a bit of a trick and you"},{"Start":"16:04.480 ","End":"16:07.645","Text":"usually look for this kind of thing if you have x is leftover."},{"Start":"16:07.645 ","End":"16:10.360","Text":"Doesn\u0027t say it may not always work."},{"Start":"16:10.360 ","End":"16:20.140","Text":"Continuing, this is equal to t cubed over 3 minus t. Finally,"},{"Start":"16:20.140 ","End":"16:23.425","Text":"we substitute what t was,"},{"Start":"16:23.425 ","End":"16:27.020","Text":"which was square root of x squared plus 1."},{"Start":"16:27.090 ","End":"16:33.925","Text":"I will write the in fractional exponent notation,"},{"Start":"16:33.925 ","End":"16:43.540","Text":"so this will be 1/3 of x squared plus 1 to the power of 3 over 2,"},{"Start":"16:43.540 ","End":"16:46.060","Text":"the 3 from here and the 2 from the square root,"},{"Start":"16:46.060 ","End":"16:52.600","Text":"3 over 2 minus x squared plus 1 to the power"},{"Start":"16:52.600 ","End":"16:59.545","Text":"of 1/2 plus c. Done for this clip."},{"Start":"16:59.545 ","End":"17:02.410","Text":"There\u0027s plenty of exercises that follow,"},{"Start":"17:02.410 ","End":"17:05.420","Text":"and do as many as you can."}],"ID":3120},{"Watched":false,"Name":"Exercise 1","Duration":"2m 7s","ChapterTopicVideoID":6678,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.080","Text":"In this exercise, we have to compute the integral of minus x over x squared plus 4,"},{"Start":"00:07.080 ","End":"00:11.670","Text":"and we\u0027re going to use the technique of integration by substitution."},{"Start":"00:11.670 ","End":"00:13.530","Text":"There are standard steps for this."},{"Start":"00:13.530 ","End":"00:16.140","Text":"First of all, we decide what to substitute."},{"Start":"00:16.140 ","End":"00:19.490","Text":"The obvious thing is to go for x squared plus 4,"},{"Start":"00:19.490 ","End":"00:26.235","Text":"so we let x squared plus 4 equal t. Then we differentiate this."},{"Start":"00:26.235 ","End":"00:29.280","Text":"On this side we get 2x."},{"Start":"00:29.280 ","End":"00:32.235","Text":"Don\u0027t forget the dx."},{"Start":"00:32.235 ","End":"00:35.760","Text":"Similarly on the other side we get 1, but it\u0027s dt."},{"Start":"00:35.760 ","End":"00:40.230","Text":"We extract dx, that\u0027s straightforward."},{"Start":"00:40.230 ","End":"00:43.304","Text":"Dx equals this over 2x,"},{"Start":"00:43.304 ","End":"00:47.610","Text":"so it\u0027s dt over 2x."},{"Start":"00:47.610 ","End":"00:52.640","Text":"At this point, I highlight both the quantity to"},{"Start":"00:52.640 ","End":"00:57.979","Text":"substitute the expression and also the dx,"},{"Start":"00:57.979 ","End":"01:02.570","Text":"because these are the 2 things that I have to replace in the original."},{"Start":"01:02.570 ","End":"01:07.010","Text":"From here, we get the integral minus x,"},{"Start":"01:07.010 ","End":"01:08.345","Text":"I don\u0027t touch it."},{"Start":"01:08.345 ","End":"01:12.000","Text":"This I can replace with t,"},{"Start":"01:12.000 ","End":"01:17.850","Text":"and the dx with dt over 2x."},{"Start":"01:17.850 ","End":"01:24.155","Text":"The x cancels, and all I am left with is minus a half."},{"Start":"01:24.155 ","End":"01:27.965","Text":"If I take the minus here with the 2 here, minus a half,"},{"Start":"01:27.965 ","End":"01:32.640","Text":"the integral of dt over t,"},{"Start":"01:32.640 ","End":"01:35.160","Text":"or 1 over t dt, if you like."},{"Start":"01:35.160 ","End":"01:38.990","Text":"The 1 over t gives us a natural logarithm,"},{"Start":"01:38.990 ","End":"01:45.335","Text":"so we have minus a half natural logarithm of t plus a constant."},{"Start":"01:45.335 ","End":"01:49.370","Text":"Finally, we substitute back from t to x."},{"Start":"01:49.370 ","End":"01:58.700","Text":"The answer is minus a half natural log of x squared plus 4 plus a constant."},{"Start":"01:58.700 ","End":"02:00.680","Text":"Here I put absolute value,"},{"Start":"02:00.680 ","End":"02:07.800","Text":"but here I don\u0027t need to because x squared plus 4 is always positive. This is the answer."}],"ID":6739},{"Watched":false,"Name":"Exercise 2","Duration":"1m 56s","ChapterTopicVideoID":6679,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.805","Text":"In this exercise, we have to compute the integral"},{"Start":"00:02.805 ","End":"00:06.405","Text":"of x times 2x squared plus 1 to the 5th,"},{"Start":"00:06.405 ","End":"00:11.385","Text":"and I\u0027m going to use the technique of integration by substitution."},{"Start":"00:11.385 ","End":"00:13.770","Text":"This goes in certain standard steps."},{"Start":"00:13.770 ","End":"00:16.395","Text":"The first step, to decide what to substitute,"},{"Start":"00:16.395 ","End":"00:19.680","Text":"the natural thing seems to be 2x squared plus 1,"},{"Start":"00:19.680 ","End":"00:24.465","Text":"and I let this equals t. Next step is to differentiate this,"},{"Start":"00:24.465 ","End":"00:26.580","Text":"so on this side, I get 4x,"},{"Start":"00:26.580 ","End":"00:29.385","Text":"on this side, I get 1,"},{"Start":"00:29.385 ","End":"00:33.415","Text":"but this is dx, and this is dt."},{"Start":"00:33.415 ","End":"00:35.790","Text":"Then I extract dx out of all this."},{"Start":"00:35.790 ","End":"00:43.405","Text":"So I just have to divide both sides by 4x, and I get dt over 4x."},{"Start":"00:43.405 ","End":"00:47.090","Text":"Now, what I do, I substitute 2 things here."},{"Start":"00:47.090 ","End":"00:49.910","Text":"I substitute on the 1 hand, dx,"},{"Start":"00:49.910 ","End":"00:53.590","Text":"on the other hand, this 2x squared plus 1."},{"Start":"00:53.590 ","End":"00:58.180","Text":"So here it\u0027s dx, and here it\u0027s 2x squared plus 1."},{"Start":"00:58.550 ","End":"01:01.205","Text":"I\u0027ll take this over here."},{"Start":"01:01.205 ","End":"01:10.115","Text":"We get the integral of x times t^5 times dx,"},{"Start":"01:10.115 ","End":"01:15.610","Text":"which is dt over 4x."},{"Start":"01:15.610 ","End":"01:21.700","Text":"The x cancels, as it often does, and all we\u0027re left with is 1/4,"},{"Start":"01:21.700 ","End":"01:28.460","Text":"which I put in front of the integral, times t^5 dt."},{"Start":"01:28.460 ","End":"01:30.860","Text":"This equals raise the power by 1,"},{"Start":"01:30.860 ","End":"01:32.810","Text":"which is 6, and divide by it."},{"Start":"01:32.810 ","End":"01:35.000","Text":"So it\u0027s 1 over 24,"},{"Start":"01:35.000 ","End":"01:41.780","Text":"which is 4 times 6 times t^6 plus a constant."},{"Start":"01:41.780 ","End":"01:46.070","Text":"Finally, I go back from t to the world of x."},{"Start":"01:46.070 ","End":"01:49.135","Text":"So we get 1 over 24,"},{"Start":"01:49.135 ","End":"01:54.195","Text":"2x squared plus 1^6,"},{"Start":"01:54.195 ","End":"01:57.370","Text":"and we are done."}],"ID":6740},{"Watched":false,"Name":"Exercise 3","Duration":"1m 30s","ChapterTopicVideoID":6680,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.130","Text":"In this exercise, I need to compute the integral of natural log of x squared over x dx."},{"Start":"00:08.130 ","End":"00:12.420","Text":"I\u0027m going to use the technique of integration by substitution."},{"Start":"00:12.420 ","End":"00:15.570","Text":"The first thing to do is to decide what to substitute,"},{"Start":"00:15.570 ","End":"00:20.040","Text":"and natural log of x seems to be the obvious choice,"},{"Start":"00:20.040 ","End":"00:23.310","Text":"I let that equals t. Now I differentiate,"},{"Start":"00:23.310 ","End":"00:29.520","Text":"so I get 1 over x dx is equal to 1 dt."},{"Start":"00:29.520 ","End":"00:31.745","Text":"If I extract dx,"},{"Start":"00:31.745 ","End":"00:37.205","Text":"I can do this by multiplying both sides by x and I get x dt."},{"Start":"00:37.205 ","End":"00:41.510","Text":"At this point, I need to substitute 2 things."},{"Start":"00:41.510 ","End":"00:47.090","Text":"I need to substitute what t was and that will go here,"},{"Start":"00:47.090 ","End":"00:50.980","Text":"and I also need to substitute dx and that is here."},{"Start":"00:50.980 ","End":"00:56.420","Text":"Continuing from here, we get the integral of"},{"Start":"00:56.420 ","End":"01:03.150","Text":"t squared over x and dx is x dt."},{"Start":"01:03.150 ","End":"01:10.680","Text":"The x cancels and all we\u0027re left with is the integral of t squared dt,"},{"Start":"01:10.850 ","End":"01:18.075","Text":"which equals 1/3 of t cubed plus a constant."},{"Start":"01:18.075 ","End":"01:21.830","Text":"Finally, we replace t back by natural log of x,"},{"Start":"01:21.830 ","End":"01:27.950","Text":"so we get 1/3 the natural log of x cubed."},{"Start":"01:27.950 ","End":"01:31.600","Text":"Done."}],"ID":6741},{"Watched":false,"Name":"Exercise 4","Duration":"1m 38s","ChapterTopicVideoID":6681,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.330","Text":"In this exercise, we have to compute the integral of e^x squared plus 4 times x."},{"Start":"00:06.330 ","End":"00:09.840","Text":"We\u0027re going to use a technique of integration by substitution."},{"Start":"00:09.840 ","End":"00:15.585","Text":"The obvious thing to substitute would be the x squared plus 4,"},{"Start":"00:15.585 ","End":"00:19.695","Text":"and we let that equal t. When we differentiate this,"},{"Start":"00:19.695 ","End":"00:26.110","Text":"we\u0027ll get 2xdx equals 1 times dt."},{"Start":"00:26.110 ","End":"00:28.529","Text":"If we extract dx,"},{"Start":"00:28.529 ","End":"00:32.710","Text":"we get that it\u0027s equal to dt over 2x."},{"Start":"00:32.990 ","End":"00:40.185","Text":"Now, there are 2 things that I want to substitute into the original integral that is dx,"},{"Start":"00:40.185 ","End":"00:42.690","Text":"which I will substitute here,"},{"Start":"00:42.690 ","End":"00:47.240","Text":"and also this expression here which I\u0027ll"},{"Start":"00:47.240 ","End":"00:52.190","Text":"substitute for t and the rest we leave alone initially."},{"Start":"00:52.190 ","End":"00:55.325","Text":"Looking back here and I\u0027m continuing over here,"},{"Start":"00:55.325 ","End":"01:01.400","Text":"I get the integral of e to the power of this thing is t,"},{"Start":"01:01.400 ","End":"01:03.230","Text":"I still have an x here,"},{"Start":"01:03.230 ","End":"01:07.435","Text":"and dx is dt over 2x."},{"Start":"01:07.435 ","End":"01:14.060","Text":"As usual, the x cancels and I\u0027ll take the 1/2 out in front so this is equal to 1/2,"},{"Start":"01:14.060 ","End":"01:24.189","Text":"the integral of e^t dt and the integral of e^t is just e^t, plus a constant."},{"Start":"01:24.189 ","End":"01:34.250","Text":"Finally, we have to go back from t to x so we just put this instead and we get 1/2,"},{"Start":"01:34.250 ","End":"01:39.660","Text":"e^x squared plus 4 plus C. We\u0027re done."}],"ID":6742},{"Watched":false,"Name":"Exercise 5","Duration":"1m 57s","ChapterTopicVideoID":6682,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.940","Text":"In this exercise, we have to compute the integral of e^x over e^x plus 1."},{"Start":"00:06.940 ","End":"00:10.575","Text":"We\u0027re going to use integration by substitution."},{"Start":"00:10.575 ","End":"00:15.270","Text":"What we\u0027re going to substitute will be the denominator here."},{"Start":"00:15.270 ","End":"00:21.150","Text":"I\u0027ll let e^x plus 1 equals t."},{"Start":"00:21.150 ","End":"00:22.320","Text":"I\u0027m thinking in my head that"},{"Start":"00:22.320 ","End":"00:26.325","Text":"the derivative of this is just e^x so that it looks like it\u0027ll work out."},{"Start":"00:26.325 ","End":"00:29.070","Text":"Anyway, we differentiate, and we get,"},{"Start":"00:29.070 ","End":"00:31.065","Text":"on the left side,"},{"Start":"00:31.065 ","End":"00:33.405","Text":"e^x, but don\u0027t forget the dx."},{"Start":"00:33.405 ","End":"00:36.810","Text":"On the right side, 1, but dt."},{"Start":"00:36.810 ","End":"00:38.670","Text":"If I extract dx from this,"},{"Start":"00:38.670 ","End":"00:47.610","Text":"I get that dx is equal to dt over e^x, dt over e^x."},{"Start":"00:47.610 ","End":"00:52.000","Text":"Now, there are 2 quantities that I want to substitute in this expression,"},{"Start":"00:52.000 ","End":"00:54.320","Text":"1 of them is e^x plus 1,"},{"Start":"00:54.320 ","End":"00:55.715","Text":"which I\u0027ll put here,"},{"Start":"00:55.715 ","End":"00:58.045","Text":"and the other 1 is the dx,"},{"Start":"00:58.045 ","End":"01:00.405","Text":"which I\u0027ll put here."},{"Start":"01:00.405 ","End":"01:07.730","Text":"Continuing over here, we get the integral of e^x."},{"Start":"01:07.730 ","End":"01:09.515","Text":"I can\u0027t replace that,"},{"Start":"01:09.515 ","End":"01:12.770","Text":"but I can replace e^x plus 1 as t,"},{"Start":"01:12.770 ","End":"01:16.720","Text":"and then dx is dt over e^x,"},{"Start":"01:16.720 ","End":"01:18.420","Text":"and, aren\u0027t we lucky?"},{"Start":"01:18.420 ","End":"01:23.955","Text":"The e^x cancels, so this becomes the integral of"},{"Start":"01:23.955 ","End":"01:33.200","Text":"dt over t. The integral of 1 over t is natural log,"},{"Start":"01:33.200 ","End":"01:40.350","Text":"so this is natural log of absolute value of t plus the constant, of course."},{"Start":"01:40.350 ","End":"01:45.335","Text":"Finally, we put t back to what it was in terms of x."},{"Start":"01:45.335 ","End":"01:51.905","Text":"This is equal to the natural log of e^x plus 1."},{"Start":"01:51.905 ","End":"01:55.459","Text":"I dropped the absolute value because this thing is positive,"},{"Start":"01:55.459 ","End":"01:57.810","Text":"and we are done."}],"ID":6743},{"Watched":false,"Name":"Exercise 6","Duration":"2m 6s","ChapterTopicVideoID":6683,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"In this exercise, we have to compute the"},{"Start":"00:02.160 ","End":"00:06.300","Text":"integral of x times the square root of x squared plus 1."},{"Start":"00:06.300 ","End":"00:09.855","Text":"We\u0027ll use the technique of integration by substitution."},{"Start":"00:09.855 ","End":"00:12.960","Text":"The first thing to do is to decide what to substitute."},{"Start":"00:12.960 ","End":"00:16.710","Text":"Actually, it\u0027s possible to substitute x squared plus 1,"},{"Start":"00:16.710 ","End":"00:20.485","Text":"but from experience, it\u0027s better to substitute it with the square root."},{"Start":"00:20.485 ","End":"00:26.010","Text":"Let\u0027s let the square root of x squared plus 1"},{"Start":"00:26.010 ","End":"00:31.500","Text":"equal t. But it would be better to get rid of the square root before we differentiate."},{"Start":"00:31.500 ","End":"00:41.189","Text":"Let\u0027s square both sides, and get that x squared plus 1 is equal to t squared,"},{"Start":"00:41.189 ","End":"00:45.570","Text":"and so 2x here and 2t here,"},{"Start":"00:45.570 ","End":"00:49.230","Text":"but let\u0027s not forget the dx and the dt."},{"Start":"00:49.230 ","End":"00:58.570","Text":"So if we extract dx, we can cancel by 2, and we\u0027re left with dx equals tdt over x."},{"Start":"01:01.070 ","End":"01:05.135","Text":"Now what I want to do is to substitute"},{"Start":"01:05.135 ","End":"01:10.400","Text":"this expression where I got it from, which was here."},{"Start":"01:10.400 ","End":"01:13.745","Text":"I also want to substitute dx."},{"Start":"01:13.745 ","End":"01:16.205","Text":"What I get is the integral."},{"Start":"01:16.205 ","End":"01:18.095","Text":"The x, I leave alone."},{"Start":"01:18.095 ","End":"01:25.405","Text":"This I said was t, and dx is tdt over x."},{"Start":"01:25.405 ","End":"01:29.310","Text":"Now x cancels."},{"Start":"01:29.310 ","End":"01:33.680","Text":"So we\u0027re left with the integral of t squared dt."},{"Start":"01:33.680 ","End":"01:40.075","Text":"The integral of t squared is 1/3 of t cubed."},{"Start":"01:40.075 ","End":"01:44.260","Text":"Finally, we have to go back from t to x."},{"Start":"01:44.260 ","End":"01:46.700","Text":"We get this thing cubed."},{"Start":"01:46.700 ","End":"01:49.594","Text":"I prefer to write it in the fractional exponents."},{"Start":"01:49.594 ","End":"01:51.930","Text":"This is x squared plus 1^1/2,"},{"Start":"01:51.930 ","End":"01:57.380","Text":"and to the power of 1/2^3 is to the power of 3/2."},{"Start":"01:57.380 ","End":"02:01.450","Text":"So we have x squared plus 1 to the power of 3/2."},{"Start":"02:01.450 ","End":"02:07.250","Text":"The 1/3 is here, and the constant, and we are done."}],"ID":6744},{"Watched":false,"Name":"Exercise 7","Duration":"2m 26s","ChapterTopicVideoID":6684,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.044","Text":"In this exercise, we have to compute the integral"},{"Start":"00:03.044 ","End":"00:06.985","Text":"of x over the square root of x squared plus 4."},{"Start":"00:06.985 ","End":"00:11.430","Text":"What we\u0027re going to do is integration by substitution."},{"Start":"00:11.430 ","End":"00:14.730","Text":"This follows a standard set of steps."},{"Start":"00:14.730 ","End":"00:18.180","Text":"The first thing is to decide what to substitute."},{"Start":"00:18.180 ","End":"00:20.565","Text":"Really there are 2 possibilities,"},{"Start":"00:20.565 ","End":"00:23.910","Text":"x squared plus 4 or square root of x squared plus 4."},{"Start":"00:23.910 ","End":"00:27.390","Text":"But by experience, it\u0027s best to substitute the whole square root."},{"Start":"00:27.390 ","End":"00:31.680","Text":"We let the square root of x squared plus 4"},{"Start":"00:31.680 ","End":"00:36.195","Text":"equal another variable t. Then we differentiate,"},{"Start":"00:36.195 ","End":"00:41.085","Text":"but first we square this in order to get rid of the square root sign."},{"Start":"00:41.085 ","End":"00:45.345","Text":"We get x squared plus 4 equals t squared,"},{"Start":"00:45.345 ","End":"00:47.100","Text":"and now we differentiate."},{"Start":"00:47.100 ","End":"00:52.680","Text":"The derivative of x squared is 2x and here it\u0027s 2t,"},{"Start":"00:52.680 ","End":"00:55.460","Text":"but we have to put a dx here"},{"Start":"00:55.460 ","End":"00:58.820","Text":"because this is the world of x and in the world of t we put a dt,"},{"Start":"00:58.820 ","End":"01:01.130","Text":"that\u0027s just the way it works."},{"Start":"01:01.130 ","End":"01:03.290","Text":"Then I extract dx,"},{"Start":"01:03.290 ","End":"01:05.405","Text":"or first I can cancel by 2."},{"Start":"01:05.405 ","End":"01:08.935","Text":"Then dx, I just divide both sides by x,"},{"Start":"01:08.935 ","End":"01:16.835","Text":"so I get that dx equals t dt over x."},{"Start":"01:16.835 ","End":"01:22.460","Text":"Now I need to do 2 substitutions instead of dx here,"},{"Start":"01:22.460 ","End":"01:24.800","Text":"I have to substitute what it is."},{"Start":"01:24.800 ","End":"01:28.250","Text":"Also the x squared plus 4,"},{"Start":"01:28.250 ","End":"01:30.465","Text":"which is what I took for t,"},{"Start":"01:30.465 ","End":"01:33.200","Text":"we also have to substitute that here."},{"Start":"01:33.200 ","End":"01:36.605","Text":"We get, I\u0027m continuing down here,"},{"Start":"01:36.605 ","End":"01:41.870","Text":"the integral of x stays as is."},{"Start":"01:41.870 ","End":"01:45.485","Text":"On the bottom we have t,"},{"Start":"01:45.485 ","End":"01:54.850","Text":"and for dx we have t dt over x."},{"Start":"01:54.850 ","End":"02:00.254","Text":"So x cancels, and it looks like t cancels also."},{"Start":"02:00.254 ","End":"02:04.400","Text":"All we\u0027re left with is the integral of dt,"},{"Start":"02:04.400 ","End":"02:07.480","Text":"which I\u0027ll write as 1 dt,"},{"Start":"02:07.480 ","End":"02:12.465","Text":"and that equals just t plus constant."},{"Start":"02:12.465 ","End":"02:19.110","Text":"Finally back to the world of x\u0027s we get the square root of x"},{"Start":"02:19.110 ","End":"02:27.400","Text":"squared plus 4 plus a constant. That\u0027s it."}],"ID":6745},{"Watched":false,"Name":"Exercise 8","Duration":"3m 25s","ChapterTopicVideoID":6685,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"In this exercise, we have to compute the integral of"},{"Start":"00:03.600 ","End":"00:07.680","Text":"x cubed times square root of x squared plus 1."},{"Start":"00:07.680 ","End":"00:10.785","Text":"We\u0027re going to use integration by substitution."},{"Start":"00:10.785 ","End":"00:13.305","Text":"Let\u0027s decide what to substitute."},{"Start":"00:13.305 ","End":"00:19.190","Text":"Probably the best is the square root of x squared plus 1."},{"Start":"00:19.190 ","End":"00:21.345","Text":"We let this equal t,"},{"Start":"00:21.345 ","End":"00:23.340","Text":"and then we differentiate."},{"Start":"00:23.340 ","End":"00:25.185","Text":"But before that, to make it easier,"},{"Start":"00:25.185 ","End":"00:27.900","Text":"we raise both sides to the power of 2,"},{"Start":"00:27.900 ","End":"00:29.325","Text":"get rid of the square root."},{"Start":"00:29.325 ","End":"00:31.590","Text":"We get that x squared plus 1,"},{"Start":"00:31.590 ","End":"00:34.455","Text":"is t squared, and now we differentiate."},{"Start":"00:34.455 ","End":"00:39.255","Text":"Here we get 2_x. Don\u0027t forget the d_x and here 2_t."},{"Start":"00:39.255 ","End":"00:43.650","Text":"Of course, with d_t. When we extract d_x,"},{"Start":"00:43.650 ","End":"00:47.460","Text":"the 2s cancel, and the x goes into the denominator."},{"Start":"00:47.460 ","End":"00:54.270","Text":"We\u0027re left with tdt over the x from here."},{"Start":"00:54.270 ","End":"00:57.600","Text":"Then I want to replace."},{"Start":"00:57.600 ","End":"00:59.945","Text":"I\u0027ll take the d_x from here,"},{"Start":"00:59.945 ","End":"01:01.760","Text":"and I want to put it here."},{"Start":"01:01.760 ","End":"01:06.510","Text":"I take this and put it here."},{"Start":"01:07.190 ","End":"01:10.310","Text":"When I continue this over here,"},{"Start":"01:10.310 ","End":"01:15.740","Text":"I get the integral now of x cubed just stays."},{"Start":"01:15.740 ","End":"01:19.075","Text":"The x squared plus 1 becomes t,"},{"Start":"01:19.075 ","End":"01:22.375","Text":"and the d_x becomes"},{"Start":"01:22.375 ","End":"01:28.000","Text":"tdt over"},{"Start":"01:28.000 ","End":"01:32.655","Text":"x. X cancels partially."},{"Start":"01:32.655 ","End":"01:38.650","Text":"What we\u0027re left with is the integral of x squared after canceling,"},{"Start":"01:38.650 ","End":"01:42.785","Text":"and t with t becomes t squared d_t."},{"Start":"01:42.785 ","End":"01:47.185","Text":"This is a bit of a problem because we were supposed to be left with only Ts,"},{"Start":"01:47.185 ","End":"01:49.705","Text":"and we also have x in here."},{"Start":"01:49.705 ","End":"01:53.515","Text":"We have to think of some trickery or"},{"Start":"01:53.515 ","End":"01:58.030","Text":"algebraic manipulation to try and get rid of the x squared."},{"Start":"01:58.030 ","End":"01:59.530","Text":"If we\u0027re not able to do that,"},{"Start":"01:59.530 ","End":"02:03.100","Text":"it means that we may have made a mistake in the calculations or"},{"Start":"02:03.100 ","End":"02:07.630","Text":"simply that this integral is not good for substitution,"},{"Start":"02:07.630 ","End":"02:09.960","Text":"or that we weren\u0027t creative enough."},{"Start":"02:09.960 ","End":"02:11.425","Text":"But I think if we look,"},{"Start":"02:11.425 ","End":"02:15.475","Text":"this line will be useful to us if we have x squared."},{"Start":"02:15.475 ","End":"02:18.190","Text":"We have what x squared plus 1 equals."},{"Start":"02:18.190 ","End":"02:22.160","Text":"Clearly, x squared is just t squared minus 1."},{"Start":"02:22.160 ","End":"02:28.795","Text":"Instead of x squared, I\u0027ll put t squared minus 1 times t squared d_t."},{"Start":"02:28.795 ","End":"02:31.795","Text":"Now everything is in terms of t, and we\u0027re fine."},{"Start":"02:31.795 ","End":"02:39.310","Text":"Just multiply out, t to the fourth minus t squared d_t."},{"Start":"02:39.310 ","End":"02:45.470","Text":"This becomes t to the 5th over 5,"},{"Start":"02:45.470 ","End":"02:48.960","Text":"minus t cubed over 3."},{"Start":"02:48.960 ","End":"02:59.150","Text":"The final step is to put it in terms of x. I\u0027ll use fractional exponents."},{"Start":"02:59.150 ","End":"03:02.480","Text":"This thing is x squared plus 1^1.5."},{"Start":"03:02.480 ","End":"03:08.555","Text":"Here we have x squared plus 1^1.5^5,"},{"Start":"03:08.555 ","End":"03:11.015","Text":"makes it 5 over 2."},{"Start":"03:11.015 ","End":"03:21.305","Text":"X squared plus 1^3^3 makes it to the power of 3 over 2, plus the constant."},{"Start":"03:21.305 ","End":"03:25.770","Text":"This is the answer. We\u0027re done."}],"ID":6746},{"Watched":false,"Name":"Exercise 9","Duration":"4m 12s","ChapterTopicVideoID":6686,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"In this exercise, we have to compute the integral of"},{"Start":"00:03.120 ","End":"00:06.870","Text":"x to the 5 times the square root of x squared plus 1."},{"Start":"00:06.870 ","End":"00:10.275","Text":"We\u0027ll use integration by substitution."},{"Start":"00:10.275 ","End":"00:13.170","Text":"First we have to decide what to substitute."},{"Start":"00:13.170 ","End":"00:19.305","Text":"From experience, you probably guessed that it\u0027s the square root of x squared plus 1,"},{"Start":"00:19.305 ","End":"00:25.055","Text":"which we let equal to say t. Before we differentiate,"},{"Start":"00:25.055 ","End":"00:26.090","Text":"to make life easier,"},{"Start":"00:26.090 ","End":"00:28.115","Text":"we square both sides."},{"Start":"00:28.115 ","End":"00:31.970","Text":"We get that x squared plus 1 equals t squared."},{"Start":"00:31.970 ","End":"00:39.220","Text":"Now we differentiate and get 2x dx equals 2t dt."},{"Start":"00:39.220 ","End":"00:41.970","Text":"Now we extract dx."},{"Start":"00:41.970 ","End":"00:43.785","Text":"So dx equals,"},{"Start":"00:43.785 ","End":"00:47.010","Text":"you can cancel the 2 and the x goes to the other side."},{"Start":"00:47.010 ","End":"00:51.790","Text":"We get t dt over x."},{"Start":"00:51.790 ","End":"00:54.845","Text":"Now I want to highlight this,"},{"Start":"00:54.845 ","End":"00:59.210","Text":"which I substituted and I\u0027m going to put it here."},{"Start":"00:59.210 ","End":"01:06.075","Text":"Also the dx, I\u0027m going to substitute here."},{"Start":"01:06.075 ","End":"01:08.975","Text":"If I continue this integral over here,"},{"Start":"01:08.975 ","End":"01:13.590","Text":"I get the integral x to the 5 just stays,"},{"Start":"01:13.590 ","End":"01:23.670","Text":"x squared plus 1 becomes t and dx becomes t dt over x."},{"Start":"01:23.670 ","End":"01:25.400","Text":"If I simplify this,"},{"Start":"01:25.400 ","End":"01:31.520","Text":"what I get is the integral x to the 5 over x is x to the 4,"},{"Start":"01:31.520 ","End":"01:36.040","Text":"t times t is t squared and dt."},{"Start":"01:36.040 ","End":"01:40.375","Text":"Now we see that we have x\u0027s here, not just t\u0027s."},{"Start":"01:40.375 ","End":"01:43.085","Text":"Maybe we made a mistake,"},{"Start":"01:43.085 ","End":"01:46.700","Text":"or maybe this is not solvable by substitution,"},{"Start":"01:46.700 ","End":"01:51.515","Text":"or maybe we just have to be creative and think how to get rid of the x to the 4."},{"Start":"01:51.515 ","End":"01:54.935","Text":"Well, I have an idea, this line here."},{"Start":"01:54.935 ","End":"01:57.170","Text":"Let me extract x squared."},{"Start":"01:57.170 ","End":"02:00.230","Text":"If I have x squared, then certainly I can have x to the 4."},{"Start":"02:00.230 ","End":"02:04.250","Text":"So x squared is t squared minus 1,"},{"Start":"02:04.250 ","End":"02:06.140","Text":"but that\u0027s x squared."},{"Start":"02:06.140 ","End":"02:07.605","Text":"I want x to the 4,"},{"Start":"02:07.605 ","End":"02:11.915","Text":"so I\u0027ll square that and that will give me x to the 4."},{"Start":"02:11.915 ","End":"02:16.410","Text":"Then I have just times t squared dt."},{"Start":"02:17.000 ","End":"02:19.200","Text":"What is this equal to?"},{"Start":"02:19.200 ","End":"02:21.765","Text":"I\u0027ll just have to do some algebra here."},{"Start":"02:21.765 ","End":"02:24.360","Text":"T squared plus 1,"},{"Start":"02:24.360 ","End":"02:33.820","Text":"this is t to the 4 minus 2t squared plus 1 times t squared dt."},{"Start":"02:33.820 ","End":"02:36.170","Text":"Just by raising this to the power of 2,"},{"Start":"02:36.170 ","End":"02:41.720","Text":"I get this and now I\u0027ll raise each of these by 2, the exponents."},{"Start":"02:41.720 ","End":"02:51.655","Text":"I get t to the 6 minus 2t to the 4 plus t squared, all this dt."},{"Start":"02:51.655 ","End":"02:55.680","Text":"They\u0027re just exponents, raise the power by 1,"},{"Start":"02:55.680 ","End":"02:58.665","Text":"it\u0027s t to the 7 over 7."},{"Start":"02:58.665 ","End":"03:01.170","Text":"I\u0027ll write it as 1/7 t to the 7."},{"Start":"03:01.170 ","End":"03:04.980","Text":"Here, raise it to 1 it\u0027s to the power of 5 now,"},{"Start":"03:04.980 ","End":"03:06.525","Text":"and divide by 5."},{"Start":"03:06.525 ","End":"03:09.615","Text":"So it\u0027s 2/5 t to the 5,"},{"Start":"03:09.615 ","End":"03:13.815","Text":"and here plus 1/3 t cubed."},{"Start":"03:13.815 ","End":"03:19.295","Text":"All we have to do now is to substitute back instead of t,"},{"Start":"03:19.295 ","End":"03:22.850","Text":"I have to put square root of x squared plus 1."},{"Start":"03:22.850 ","End":"03:27.695","Text":"The square root of x squared plus 1 is x squared plus 1 to the power of a 1/2."},{"Start":"03:27.695 ","End":"03:35.215","Text":"So x squared plus 1 to the power of 7,"},{"Start":"03:35.215 ","End":"03:38.025","Text":"gives me to the power of a 7 over 2."},{"Start":"03:38.025 ","End":"03:44.370","Text":"Similarly, minus 2/5 x squared plus 1 to the 5 over 2,"},{"Start":"03:44.370 ","End":"03:54.890","Text":"and plus 1/3, x squared plus 1 to the power of 3 over 2 plus the constant."},{"Start":"03:54.890 ","End":"03:57.710","Text":"Probably this could be opened out and simplified."},{"Start":"03:57.710 ","End":"03:59.330","Text":"If we took t out here,"},{"Start":"03:59.330 ","End":"04:02.540","Text":"then t squared is x squared plus 1."},{"Start":"04:02.540 ","End":"04:03.710","Text":"We could do some algebra,"},{"Start":"04:03.710 ","End":"04:08.990","Text":"but the purpose here is the integration by substitution and not the algebra."},{"Start":"04:08.990 ","End":"04:13.440","Text":"Leave this as the answer, and we\u0027re done."}],"ID":6747},{"Watched":false,"Name":"Exercise 10","Duration":"2m 46s","ChapterTopicVideoID":6687,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.615","Text":"In this exercise, we need to compute the integral"},{"Start":"00:03.615 ","End":"00:07.875","Text":"of x cubed over square root of x squared plus 4."},{"Start":"00:07.875 ","End":"00:11.475","Text":"We\u0027ll use a technique of integration by substitution."},{"Start":"00:11.475 ","End":"00:16.005","Text":"The first thing is to decide what to substitute and"},{"Start":"00:16.005 ","End":"00:22.845","Text":"the obvious choice is the square root of x squared plus 4."},{"Start":"00:22.845 ","End":"00:26.250","Text":"Let this equal t, want to differentiate."},{"Start":"00:26.250 ","End":"00:28.020","Text":"But before that, to make life easier,"},{"Start":"00:28.020 ","End":"00:34.275","Text":"we\u0027ll square both sides and get x squared plus 4 equals t squared."},{"Start":"00:34.275 ","End":"00:41.175","Text":"Now we can say that 2x dx is equal to 2t dt."},{"Start":"00:41.175 ","End":"00:45.410","Text":"If we extract dx, the 2 cancels,"},{"Start":"00:45.410 ","End":"00:46.680","Text":"the x goes over there,"},{"Start":"00:46.680 ","End":"00:51.230","Text":"so we get t dt over x."},{"Start":"00:51.230 ","End":"00:56.075","Text":"Now I want to substitute the dx"},{"Start":"00:56.075 ","End":"01:02.645","Text":"here and also the square root of x squared plus 4."},{"Start":"01:02.645 ","End":"01:05.480","Text":"I want to put that here."},{"Start":"01:05.480 ","End":"01:14.000","Text":"Now this integral becomes the integral of x cubed"},{"Start":"01:14.000 ","End":"01:23.615","Text":"over t and dx is t dt over x."},{"Start":"01:23.615 ","End":"01:27.305","Text":"Let\u0027s see, x cancels with x cubed,"},{"Start":"01:27.305 ","End":"01:30.875","Text":"and that gives us x squared,"},{"Start":"01:30.875 ","End":"01:33.165","Text":"t cancels with t,"},{"Start":"01:33.165 ","End":"01:36.670","Text":"so we\u0027re left with x squared dt."},{"Start":"01:36.980 ","End":"01:42.935","Text":"Here we apparently have a problem because we want all ts and here we have xs."},{"Start":"01:42.935 ","End":"01:45.500","Text":"If we haven\u0027t made a mistake,"},{"Start":"01:45.500 ","End":"01:48.320","Text":"it still might be that we can\u0027t do it by substitution,"},{"Start":"01:48.320 ","End":"01:53.690","Text":"but the usual thing is to try and somehow convert this x expression to a t expression."},{"Start":"01:53.690 ","End":"01:56.585","Text":"The line to look at would be this 1."},{"Start":"01:56.585 ","End":"02:01.895","Text":"Because we can see that x squared is just t squared minus 4."},{"Start":"02:01.895 ","End":"02:08.870","Text":"We can write this as the integral of t squared minus 4"},{"Start":"02:08.870 ","End":"02:18.800","Text":"dt and this equals 1/3 of t cubed minus 4t plus a constant."},{"Start":"02:18.800 ","End":"02:23.825","Text":"Finally, don\u0027t forget to convert back from t to x,"},{"Start":"02:23.825 ","End":"02:26.870","Text":"so we get 1/3."},{"Start":"02:26.870 ","End":"02:33.830","Text":"Now t is x squared plus 4 to the power of a 1/2."},{"Start":"02:33.830 ","End":"02:39.090","Text":"But with the 3, it\u0027s 3 1/2 minus 4."},{"Start":"02:39.090 ","End":"02:44.510","Text":"To be consistent, I\u0027ll also write this as x squared plus 4 to the power of a 1/2,"},{"Start":"02:44.510 ","End":"02:47.730","Text":"and we are done."}],"ID":6748},{"Watched":false,"Name":"Exercise 11","Duration":"3m 8s","ChapterTopicVideoID":6688,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.255","Text":"In this exercise, we have to compute the integral"},{"Start":"00:03.255 ","End":"00:08.385","Text":"of x^5 over the square root of x cubed plus 1."},{"Start":"00:08.385 ","End":"00:11.580","Text":"But first, we\u0027re going to do this with substitution,"},{"Start":"00:11.580 ","End":"00:14.040","Text":"that is the technique of integration by substitution."},{"Start":"00:14.040 ","End":"00:17.040","Text":"The first thing is to decide what to substitute."},{"Start":"00:17.040 ","End":"00:22.680","Text":"The obvious choice is the square root of x cubed plus 1."},{"Start":"00:22.680 ","End":"00:24.030","Text":"Let that equal t,"},{"Start":"00:24.030 ","End":"00:26.384","Text":"is our favorite variable."},{"Start":"00:26.384 ","End":"00:32.985","Text":"Before we differentiate, we square both sides to make life easier."},{"Start":"00:32.985 ","End":"00:37.755","Text":"X cubed plus 1 equals t squared."},{"Start":"00:37.755 ","End":"00:43.680","Text":"Then differentiating, we get 3x squared on this side, dx."},{"Start":"00:43.680 ","End":"00:47.200","Text":"On the other side 2t goes with dt."},{"Start":"00:47.240 ","End":"00:49.965","Text":"Now we extract dx,"},{"Start":"00:49.965 ","End":"00:51.465","Text":"little bit of algebra,"},{"Start":"00:51.465 ","End":"01:00.240","Text":"and we can see that this is equal to 2t dt over 3x squared."},{"Start":"01:00.240 ","End":"01:05.270","Text":"What I want to do now is to substitute this here,"},{"Start":"01:05.270 ","End":"01:11.415","Text":"and I want to substitute the dx here,"},{"Start":"01:11.415 ","End":"01:13.995","Text":"and let\u0027s see what we get."},{"Start":"01:13.995 ","End":"01:15.805","Text":"Continuing this over here,"},{"Start":"01:15.805 ","End":"01:21.599","Text":"we got the integral of x^5, I don\u0027t touch."},{"Start":"01:21.599 ","End":"01:24.765","Text":"This becomes t from here,"},{"Start":"01:24.765 ","End":"01:34.410","Text":"and this becomes 2t dt over 3x squared."},{"Start":"01:34.410 ","End":"01:37.470","Text":"Let\u0027s see, t cancels with t,"},{"Start":"01:37.470 ","End":"01:38.730","Text":"so all the t\u0027s are gone,"},{"Start":"01:38.730 ","End":"01:39.950","Text":"but the x\u0027s are still here."},{"Start":"01:39.950 ","End":"01:43.400","Text":"Let me put the 2/3 in front of the integral."},{"Start":"01:43.400 ","End":"01:46.850","Text":"What we\u0027re left with is x^5 over x squared,"},{"Start":"01:46.850 ","End":"01:50.810","Text":"which is x cubed dt."},{"Start":"01:50.810 ","End":"01:53.705","Text":"I don\u0027t want the x\u0027s, I want t\u0027s."},{"Start":"01:53.705 ","End":"01:55.970","Text":"I haven\u0027t made a mistake,"},{"Start":"01:55.970 ","End":"01:58.370","Text":"and if it\u0027s possible to do this with substitution,"},{"Start":"01:58.370 ","End":"02:00.710","Text":"I have to think of something clever."},{"Start":"02:00.710 ","End":"02:04.610","Text":"You\u0027ve probably seen these tricks before. X cubed,"},{"Start":"02:04.610 ","End":"02:08.210","Text":"if you look here, is exactly t squared minus 1."},{"Start":"02:08.210 ","End":"02:15.610","Text":"I\u0027ve got 2/3 times the integral of t squared minus 1 dt,"},{"Start":"02:15.610 ","End":"02:21.685","Text":"and this equals the integral of t squared is t cubed over 3."},{"Start":"02:21.685 ","End":"02:25.065","Text":"Let\u0027s see, we have 2/9 t cubed,"},{"Start":"02:25.065 ","End":"02:27.170","Text":"minus 1 becomes minus t,"},{"Start":"02:27.170 ","End":"02:31.565","Text":"so minus 2/3 t plus a constant."},{"Start":"02:31.565 ","End":"02:33.635","Text":"The final step, of course,"},{"Start":"02:33.635 ","End":"02:40.520","Text":"is just to go back to the world of x from t. I put the square root of x cubed plus 1,"},{"Start":"02:40.520 ","End":"02:45.500","Text":"and I\u0027ll use the exponent notation for roots,"},{"Start":"02:45.500 ","End":"02:52.730","Text":"so this will be t will be x cubed plus 1^1/2."},{"Start":"02:52.730 ","End":"02:55.865","Text":"But because of the 3, it\u0027s not a half, it\u0027s 3 over 2,"},{"Start":"02:55.865 ","End":"03:00.320","Text":"and the 2/9 is here, 2/3, this time,"},{"Start":"03:00.320 ","End":"03:02.590","Text":"x cubed plus 1^1/2,"},{"Start":"03:02.590 ","End":"03:05.295","Text":"and the constant here, of course."},{"Start":"03:05.295 ","End":"03:09.010","Text":"This is the answer, we\u0027re done."}],"ID":6749},{"Watched":false,"Name":"Exercise 12","Duration":"3m 31s","ChapterTopicVideoID":6689,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we have to compute the integral of"},{"Start":"00:03.630 ","End":"00:08.685","Text":"the fourth root of x squared plus 1 times x cubed dx."},{"Start":"00:08.685 ","End":"00:12.225","Text":"We\u0027re going to use the technique of integration by substitution."},{"Start":"00:12.225 ","End":"00:14.595","Text":"We should decide what to substitute."},{"Start":"00:14.595 ","End":"00:19.590","Text":"By experience, it\u0027s the fourth root of x squared plus 1."},{"Start":"00:19.590 ","End":"00:24.585","Text":"We let this equal t. Next step is to differentiate."},{"Start":"00:24.585 ","End":"00:26.790","Text":"But before that, to make life easier,"},{"Start":"00:26.790 ","End":"00:28.140","Text":"we raise to the power of 4."},{"Start":"00:28.140 ","End":"00:35.640","Text":"Get rid of this root this radical and we get x squared plus 1 is equal to t to the 4th."},{"Start":"00:35.640 ","End":"00:36.915","Text":"Now we differentiate."},{"Start":"00:36.915 ","End":"00:43.790","Text":"We get that 2x dx is equal to 4t cubed dt."},{"Start":"00:43.790 ","End":"00:46.175","Text":"If we extract dx,"},{"Start":"00:46.175 ","End":"00:49.175","Text":"what we get is simply dividing by 2x."},{"Start":"00:49.175 ","End":"00:56.220","Text":"The 2 with the 4 makes it just 2t cubed over xdt."},{"Start":"00:57.460 ","End":"01:01.760","Text":"Now, what I want to do is a bit of substituting."},{"Start":"01:01.760 ","End":"01:05.840","Text":"What I want to do is take the dx and put it here."},{"Start":"01:05.840 ","End":"01:11.940","Text":"I also want to take this fourth root and put it here."},{"Start":"01:11.940 ","End":"01:18.485","Text":"What we get if we continue this integral over here is the integral of"},{"Start":"01:18.485 ","End":"01:23.045","Text":"t times x cubed dx"},{"Start":"01:23.045 ","End":"01:31.390","Text":"is 2t cubed dt over x."},{"Start":"01:31.390 ","End":"01:33.670","Text":"Simplifying this a bit,"},{"Start":"01:33.670 ","End":"01:37.870","Text":"x cubed over x is x squared."},{"Start":"01:37.870 ","End":"01:46.975","Text":"The 2 comes out in front and t with t cubed gives t to the 4th and we still have a dt."},{"Start":"01:46.975 ","End":"01:51.490","Text":"The problem here is that we want all t\u0027s and we also have"},{"Start":"01:51.490 ","End":"01:55.780","Text":"x\u0027s and probably seen this trick before."},{"Start":"01:55.780 ","End":"01:59.605","Text":"I can see from this line that x squared is not hard to get."},{"Start":"01:59.605 ","End":"02:02.410","Text":"It\u0027s just t to the 4th minus 1."},{"Start":"02:02.410 ","End":"02:12.880","Text":"This becomes 2 times the integral of t to the 4th minus 1 times t to the 4th."},{"Start":"02:12.880 ","End":"02:18.165","Text":"This equals t to the 4th times t to the 4th is t to the 8th."},{"Start":"02:18.165 ","End":"02:20.150","Text":"I\u0027ll just write them separately."},{"Start":"02:20.150 ","End":"02:22.345","Text":"I\u0027ll put the 2 back in."},{"Start":"02:22.345 ","End":"02:28.160","Text":"2t to the 8th and then minus t to the 4th so it\u0027s minus 2t to the 4th."},{"Start":"02:28.890 ","End":"02:32.600","Text":"Oh, I forgot the dt here."},{"Start":"02:33.950 ","End":"02:41.560","Text":"Now I can see that this is 2 over 9t to the 9th"},{"Start":"02:41.990 ","End":"02:51.415","Text":"and this will be 2 over 5t to the 5th plus a constant."},{"Start":"02:51.415 ","End":"02:56.315","Text":"The last thing we have to do is to convert back"},{"Start":"02:56.315 ","End":"03:01.205","Text":"from the world of t to the world of x and as usual,"},{"Start":"03:01.205 ","End":"03:04.940","Text":"I\u0027ll take the power of a half rather than square root."},{"Start":"03:04.940 ","End":"03:07.055","Text":"I have 2/9."},{"Start":"03:07.055 ","End":"03:15.785","Text":"Now I have x squared plus 1 to the power of a 1/2 but to the power of 9 makes it 9 over 2."},{"Start":"03:15.785 ","End":"03:19.310","Text":"Similarly here we have 2/5 x squared"},{"Start":"03:19.310 ","End":"03:25.675","Text":"plus 1 to the power of 5 over 2 and we still have this constant."},{"Start":"03:25.675 ","End":"03:31.800","Text":"Here we are with all x\u0027s and this is the answer to the integral. We\u0027re done."}],"ID":6750},{"Watched":false,"Name":"Exercise 13","Duration":"3m 44s","ChapterTopicVideoID":6690,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we need to compute the integral of"},{"Start":"00:03.630 ","End":"00:07.545","Text":"x cubed over the cube root of x squared plus 4."},{"Start":"00:07.545 ","End":"00:11.595","Text":"We\u0027re going to use a technique of integration by substitution."},{"Start":"00:11.595 ","End":"00:14.640","Text":"The first step is to decide what to substitute."},{"Start":"00:14.640 ","End":"00:21.480","Text":"By experience, this will be the cube root so we let the cube root of this expression,"},{"Start":"00:21.480 ","End":"00:25.290","Text":"x squared plus 4 be equal to t,"},{"Start":"00:25.290 ","End":"00:28.390","Text":"our favorite variable for substitution."},{"Start":"00:28.390 ","End":"00:30.890","Text":"Before we differentiate, we want to make life"},{"Start":"00:30.890 ","End":"00:35.090","Text":"easier and let\u0027s raise both sides to the power of 3."},{"Start":"00:35.090 ","End":"00:39.650","Text":"We get that x squared plus 4 is equal to t cubed."},{"Start":"00:39.650 ","End":"00:47.750","Text":"Now we differentiate and we get that 2x dx is equal to 3t squared dt."},{"Start":"00:47.750 ","End":"00:55.700","Text":"What we want is just dx to isolate it and this is equal to this thing"},{"Start":"00:55.700 ","End":"01:03.705","Text":"over 2x so it\u0027s 3t squared dt over 2x."},{"Start":"01:03.705 ","End":"01:10.595","Text":"Now what I want to do is to take this thing and substitute it here,"},{"Start":"01:10.595 ","End":"01:16.815","Text":"and also to take dx and substitute that here."},{"Start":"01:16.815 ","End":"01:20.720","Text":"What we get if we continue from this integral,"},{"Start":"01:20.720 ","End":"01:25.460","Text":"is we get the integral x cubed just stays,"},{"Start":"01:25.460 ","End":"01:32.325","Text":"then cube root of x squared plus 4 is t from here,"},{"Start":"01:32.325 ","End":"01:41.790","Text":"and then dx is 3t squared dt over 2x."},{"Start":"01:41.790 ","End":"01:45.955","Text":"Let\u0027s see what\u0027s left after we do all this cancellation."},{"Start":"01:45.955 ","End":"01:47.475","Text":"We can get the integral."},{"Start":"01:47.475 ","End":"01:49.880","Text":"Numbers can come out in front of the integral,"},{"Start":"01:49.880 ","End":"01:52.490","Text":"so that\u0027s 3 over 2 here."},{"Start":"01:52.490 ","End":"01:56.405","Text":"X cubed over x is x squared."},{"Start":"01:56.405 ","End":"02:01.590","Text":"T squared over t is just t and dt."},{"Start":"02:01.590 ","End":"02:05.310","Text":"The trouble is that we have x here,"},{"Start":"02:05.310 ","End":"02:08.060","Text":"we want to have an expression only in t,"},{"Start":"02:08.060 ","End":"02:10.790","Text":"and you\u0027ve probably seen this trick before."},{"Start":"02:10.790 ","End":"02:12.260","Text":"We can get x squared,"},{"Start":"02:12.260 ","End":"02:13.760","Text":"if we look at this line here,"},{"Start":"02:13.760 ","End":"02:17.425","Text":"x squared is equal to t cubed minus 4."},{"Start":"02:17.425 ","End":"02:27.545","Text":"We get now 3 over 2 integral of t cubed minus 4 times t dt."},{"Start":"02:27.545 ","End":"02:30.655","Text":"Still a little bit of algebra here."},{"Start":"02:30.655 ","End":"02:38.715","Text":"Integral of t to the 4th minus 4t dt,"},{"Start":"02:38.715 ","End":"02:42.870","Text":"which equals 3 over 2,"},{"Start":"02:42.870 ","End":"02:48.630","Text":"times t to the 5th over 5"},{"Start":"02:48.630 ","End":"02:55.830","Text":"minus 4t squared over 2 plus a constant."},{"Start":"02:55.830 ","End":"02:57.675","Text":"Lightly simplify this."},{"Start":"02:57.675 ","End":"03:06.330","Text":"This is 3/10 t to the 5 minus 3 over 2 times 4 over 2 is just"},{"Start":"03:06.330 ","End":"03:15.380","Text":"3t squared plus C. All we have to do now is replace t by this thing here,"},{"Start":"03:15.380 ","End":"03:17.525","Text":"what we substituted and we\u0027ll get,"},{"Start":"03:17.525 ","End":"03:19.640","Text":"but writing to the power of a 1/3,"},{"Start":"03:19.640 ","End":"03:24.665","Text":"instead, we get t is x squared plus 4."},{"Start":"03:24.665 ","End":"03:26.660","Text":"Now it\u0027s to the power of a 1/3,"},{"Start":"03:26.660 ","End":"03:28.220","Text":"but also to the power of 5,"},{"Start":"03:28.220 ","End":"03:30.790","Text":"so to the power of 5 over 3."},{"Start":"03:30.790 ","End":"03:35.390","Text":"Then the next exponent will be to the power of a 1/3,"},{"Start":"03:35.390 ","End":"03:38.195","Text":"to the power of 2, which will be to the power of 2/3,"},{"Start":"03:38.195 ","End":"03:41.300","Text":"and the C we\u0027ve been dragging around everywhere."},{"Start":"03:41.300 ","End":"03:44.760","Text":"This is the answer and we are done."}],"ID":6751},{"Watched":false,"Name":"Exercise 14","Duration":"2m 15s","ChapterTopicVideoID":6691,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this exercise, we have to compute the integral of"},{"Start":"00:03.900 ","End":"00:08.175","Text":"1 over x times natural log of x to the 4th."},{"Start":"00:08.175 ","End":"00:13.530","Text":"I\u0027m going to use the method of integration by substitution."},{"Start":"00:13.530 ","End":"00:16.185","Text":"What we\u0027re going to substitute is fairly clear,"},{"Start":"00:16.185 ","End":"00:18.895","Text":"it\u0027s going to be the natural log of x,"},{"Start":"00:18.895 ","End":"00:24.060","Text":"and we\u0027re going to let that equal t. Then we differentiate,"},{"Start":"00:24.060 ","End":"00:26.415","Text":"so we get 1 over x,"},{"Start":"00:26.415 ","End":"00:30.750","Text":"dx is equal to 1 dt."},{"Start":"00:30.750 ","End":"00:33.405","Text":"I want to know how much dx is,"},{"Start":"00:33.405 ","End":"00:38.700","Text":"so dx is just equal to x dt."},{"Start":"00:38.700 ","End":"00:41.510","Text":"Now, I want to substitute some things,"},{"Start":"00:41.510 ","End":"00:48.415","Text":"I\u0027m going to take the natural log of x and substitute that here."},{"Start":"00:48.415 ","End":"00:56.015","Text":"I\u0027m also going to take the dx from here and substitute that here."},{"Start":"00:56.015 ","End":"00:58.175","Text":"Let\u0027s see what we get."},{"Start":"00:58.175 ","End":"01:00.109","Text":"Continuing from this integral,"},{"Start":"01:00.109 ","End":"01:06.030","Text":"we have now the integral of 1 over the x,"},{"Start":"01:06.030 ","End":"01:07.205","Text":"so I leave alone,"},{"Start":"01:07.205 ","End":"01:10.325","Text":"natural log of x is t to the 4th,"},{"Start":"01:10.325 ","End":"01:13.415","Text":"and the dx is x dt."},{"Start":"01:13.415 ","End":"01:18.030","Text":"I can just put it here as x dt."},{"Start":"01:18.620 ","End":"01:20.740","Text":"What I\u0027m left with,"},{"Start":"01:20.740 ","End":"01:28.675","Text":"the x cancels is that is this is equal to the integral of 1 over t to the 4th,"},{"Start":"01:28.675 ","End":"01:33.185","Text":"so it\u0027s t to the minus 4 dt."},{"Start":"01:33.185 ","End":"01:44.830","Text":"This equals t to the minus 3 over minus 3 plus a constant."},{"Start":"01:44.870 ","End":"01:55.725","Text":"I can write this slightly more simply as minus 1 over 3t cubed,"},{"Start":"01:55.725 ","End":"02:00.620","Text":"and finally, revert back from t to natural log of x,"},{"Start":"02:00.620 ","End":"02:08.490","Text":"so it\u0027s minus 1 over 3 times natural log"},{"Start":"02:08.490 ","End":"02:13.140","Text":"of x cubed plus a constant,"},{"Start":"02:13.140 ","End":"02:16.150","Text":"and this is the answer."}],"ID":6752},{"Watched":false,"Name":"Exercise 15","Duration":"2m 46s","ChapterTopicVideoID":6692,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"Here we have to compute the integral of e to the x squared times"},{"Start":"00:03.840 ","End":"00:10.500","Text":"x cubed dx and we\u0027ll do this using the technique of integration by substitution."},{"Start":"00:10.500 ","End":"00:14.310","Text":"The first thing to do is to decide what to substitute."},{"Start":"00:14.310 ","End":"00:21.170","Text":"I think we\u0027ll take x squared as the thing to substitute and we let it equal"},{"Start":"00:21.170 ","End":"00:29.995","Text":"t. Now we differentiate and we get 2x dx equals 1dt."},{"Start":"00:29.995 ","End":"00:33.090","Text":"What we\u0027re interested is dx."},{"Start":"00:33.090 ","End":"00:37.860","Text":"dx is equal to dt over 2x."},{"Start":"00:37.860 ","End":"00:41.555","Text":"Now what we\u0027re going do is substitute some things,"},{"Start":"00:41.555 ","End":"00:49.785","Text":"we\u0027ll put dx here and we\u0027ll put the x squared here."},{"Start":"00:49.785 ","End":"00:51.845","Text":"Let\u0027s see what we get."},{"Start":"00:51.845 ","End":"00:55.055","Text":"If we continue with this integral over here,"},{"Start":"00:55.055 ","End":"01:00.645","Text":"we get the integral of e^t times"},{"Start":"01:00.645 ","End":"01:07.080","Text":"x cubed times dx which is dt over 2x."},{"Start":"01:07.080 ","End":"01:08.775","Text":"This comes out to,"},{"Start":"01:08.775 ","End":"01:10.920","Text":"we take the half upfront."},{"Start":"01:10.920 ","End":"01:16.785","Text":"e to the t, and the x with x cubed makes it x squared dt."},{"Start":"01:16.785 ","End":"01:20.740","Text":"At first, it looks not good because we want to have"},{"Start":"01:20.740 ","End":"01:24.340","Text":"all just t and we don\u0027t want x but then we quickly"},{"Start":"01:24.340 ","End":"01:28.480","Text":"realized that we could substitute again because x squared"},{"Start":"01:28.480 ","End":"01:32.800","Text":"is equal to t. It\u0027s a good job we notice that,"},{"Start":"01:32.800 ","End":"01:36.130","Text":"otherwise, you wouldn\u0027t be able to continue but when you have x leftover,"},{"Start":"01:36.130 ","End":"01:40.330","Text":"you have to start thinking and seeing if you can convert it to t after all."},{"Start":"01:40.330 ","End":"01:48.170","Text":"It\u0027s 1.5 times the integral of t e^t."},{"Start":"01:48.170 ","End":"01:50.455","Text":"I wrote it in front, I prefer it."},{"Start":"01:50.455 ","End":"01:54.350","Text":"This is done using integration by parts,"},{"Start":"01:54.350 ","End":"01:55.545","Text":"and I\u0027m not going to do that here,"},{"Start":"01:55.545 ","End":"01:57.335","Text":"I\u0027ll just give you the answer."},{"Start":"01:57.335 ","End":"02:03.820","Text":"The half stays a half but this integral becomes t minus 1e^t."},{"Start":"02:04.390 ","End":"02:08.150","Text":"I would just recommend differentiating"},{"Start":"02:08.150 ","End":"02:11.345","Text":"this if you\u0027re not sure that this is the right thing."},{"Start":"02:11.345 ","End":"02:15.320","Text":"As a product, you\u0027ll get t minus 1,"},{"Start":"02:15.320 ","End":"02:19.965","Text":"not differentiated e to the t plus another 1e^t."},{"Start":"02:19.965 ","End":"02:22.490","Text":"The t minus 1 plus 1 will give you t. Anyway,"},{"Start":"02:22.490 ","End":"02:27.295","Text":"you can check that but this is how I got it by parts."},{"Start":"02:27.295 ","End":"02:33.660","Text":"The final thing to do is to convert back from t to put x squared."},{"Start":"02:33.660 ","End":"02:39.890","Text":"The answer is 1.5x squared minus"},{"Start":"02:39.890 ","End":"02:47.590","Text":"1 e^x squared plus the constant and that\u0027s the answer."}],"ID":6753},{"Watched":false,"Name":"Exercise 16","Duration":"3m 20s","ChapterTopicVideoID":6693,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.725","Text":"Here we have to compute the integral of x to the 7th over 1 minus x to the 4th squared,"},{"Start":"00:07.725 ","End":"00:10.590","Text":"and we\u0027re going to do it by substitution."},{"Start":"00:10.590 ","End":"00:13.215","Text":"So the first question is what to substitute?"},{"Start":"00:13.215 ","End":"00:17.130","Text":"It\u0027s partly an art or some intuition there and experience."},{"Start":"00:17.130 ","End":"00:20.650","Text":"I would say go for 1 minus x to the 4th."},{"Start":"00:20.650 ","End":"00:26.280","Text":"1 minus x to the 4th will let equal to t and then we\u0027ll"},{"Start":"00:26.280 ","End":"00:33.520","Text":"differentiate this and get minus 4x cubed dx equals 1dt."},{"Start":"00:33.650 ","End":"00:36.374","Text":"If we extract dx,"},{"Start":"00:36.374 ","End":"00:38.950","Text":"this will equal minus"},{"Start":"00:38.950 ","End":"00:43.875","Text":"dt over 4x cubed"},{"Start":"00:43.875 ","End":"00:51.285","Text":"and what I want to do now is to put dx here."},{"Start":"00:51.285 ","End":"00:58.595","Text":"I want to put the 1 minus x to the 4th here, so substituting those,"},{"Start":"00:58.595 ","End":"01:08.720","Text":"what we\u0027ll get is the integral of x to the 7th and then dx is minus I will put in front."},{"Start":"01:08.720 ","End":"01:16.310","Text":"So dt over 4x cubed here,"},{"Start":"01:16.310 ","End":"01:18.980","Text":"but I still haven\u0027t finished with this bit,"},{"Start":"01:18.980 ","End":"01:21.890","Text":"and this would be t squared."},{"Start":"01:21.890 ","End":"01:25.100","Text":"After a bit of canceling,"},{"Start":"01:25.100 ","End":"01:30.330","Text":"what we\u0027ll get is minus 1/4 the"},{"Start":"01:30.330 ","End":"01:38.760","Text":"integral of x to the 4th over t squared dt."},{"Start":"01:38.760 ","End":"01:44.795","Text":"This is not quite good for us because we don\u0027t have all t,"},{"Start":"01:44.795 ","End":"01:47.930","Text":"we have this expression involving x."},{"Start":"01:47.930 ","End":"01:54.679","Text":"So we look for some way to convert this to t and if we look at this equation,"},{"Start":"01:54.679 ","End":"02:01.055","Text":"it\u0027s easy to see that x to the 4th is 1 minus t. So minus 1/4"},{"Start":"02:01.055 ","End":"02:09.005","Text":"integral of 1 minus t dt over t squared."},{"Start":"02:09.005 ","End":"02:14.230","Text":"I would just break this up and say it\u0027s minus a 1/4."},{"Start":"02:14.230 ","End":"02:18.020","Text":"The integral, using the distributive rule for fractions,"},{"Start":"02:18.020 ","End":"02:22.070","Text":"a minus b over c is a over c minus b over c. So we"},{"Start":"02:22.070 ","End":"02:28.140","Text":"have 1 over t squared minus 1 over t dt."},{"Start":"02:28.720 ","End":"02:32.885","Text":"The integral of 1 over t squared,"},{"Start":"02:32.885 ","End":"02:34.475","Text":"it\u0027s easy to check,"},{"Start":"02:34.475 ","End":"02:42.350","Text":"is minus 1 over t. Just differentiate that or t to the minus 2 becomes t to the minus 1"},{"Start":"02:42.350 ","End":"02:51.220","Text":"over minus 1 and 1 over t is natural log of t plus a constant."},{"Start":"02:51.220 ","End":"02:57.980","Text":"Finally, I\u0027m going to substitute instead of t 1 minus x to the 4th."},{"Start":"02:57.980 ","End":"03:03.350","Text":"So also I want to make everything plus this minus can go with this minus and this minus."},{"Start":"03:03.350 ","End":"03:05.345","Text":"So we get 1/4,"},{"Start":"03:05.345 ","End":"03:10.930","Text":"1 over t is 1 minus x to the 4th"},{"Start":"03:10.930 ","End":"03:17.375","Text":"and plus natural log of 1 minus x to the 4th."},{"Start":"03:17.375 ","End":"03:21.720","Text":"That\u0027s the answer, except for the constant."}],"ID":6754},{"Watched":false,"Name":"Exercise 17","Duration":"3m 51s","ChapterTopicVideoID":6694,"CourseChapterTopicPlaylistID":2676,"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":"In this exercise, we have to compute"},{"Start":"00:02.310 ","End":"00:06.960","Text":"the integral of the square root of 1 plus e^2x."},{"Start":"00:06.960 ","End":"00:08.670","Text":"I\u0027m going to use the technique"},{"Start":"00:08.670 ","End":"00:11.025","Text":"of integration by substitution."},{"Start":"00:11.025 ","End":"00:12.900","Text":"What I suggest substituting"},{"Start":"00:12.900 ","End":"00:14.955","Text":"is the whole square root."},{"Start":"00:14.955 ","End":"00:16.080","Text":"In other words,"},{"Start":"00:16.080 ","End":"00:19.160","Text":"the square root of 1 plus e^2x,"},{"Start":"00:19.160 ","End":"00:22.740","Text":"we\u0027ll assign to the variable t,"},{"Start":"00:22.740 ","End":"00:24.510","Text":"we\u0027ll differentiate,"},{"Start":"00:24.510 ","End":"00:26.820","Text":"but not before we\u0027ve squared both sides"},{"Start":"00:26.820 ","End":"00:29.010","Text":"to get rid of that nasty square root."},{"Start":"00:29.010 ","End":"00:34.245","Text":"We get that 1 plus e^2x is equal to t squared."},{"Start":"00:34.245 ","End":"00:36.320","Text":"Differentiating, we get 2e^2x"},{"Start":"00:36.320 ","End":"00:40.230","Text":"and don\u0027t forget the dx,"},{"Start":"00:40.230 ","End":"00:44.010","Text":"and also 2t and don\u0027t forget the dt."},{"Start":"00:44.010 ","End":"00:45.870","Text":"2 cancels."},{"Start":"00:45.870 ","End":"00:47.675","Text":"What I want is dx,"},{"Start":"00:47.675 ","End":"01:01.285","Text":"and this is going to equal tdt over e^2x."},{"Start":"01:01.285 ","End":"01:02.880","Text":"Now, what I want to do"},{"Start":"01:02.880 ","End":"01:08.670","Text":"is to substitute this bit here."},{"Start":"01:08.780 ","End":"01:15.220","Text":"I want to substitute the dx here."},{"Start":"01:15.320 ","End":"01:18.080","Text":"What we get with this integral,"},{"Start":"01:18.080 ","End":"01:19.775","Text":"if we continue over here,"},{"Start":"01:19.775 ","End":"01:22.895","Text":"is the integral of this bit,"},{"Start":"01:22.895 ","End":"01:28.770","Text":"which is t and the dx is tdt,"},{"Start":"01:29.770 ","End":"01:36.680","Text":"all this over e^2x."},{"Start":"01:36.680 ","End":"01:38.180","Text":"It doesn\u0027t look good at first,"},{"Start":"01:38.180 ","End":"01:39.770","Text":"because I have x in here,"},{"Start":"01:39.770 ","End":"01:42.460","Text":"and I want it to be all t,"},{"Start":"01:42.460 ","End":"01:45.890","Text":"so I look for some trick or substitution."},{"Start":"01:45.890 ","End":"01:50.150","Text":"I can see that in this line here,"},{"Start":"01:50.150 ","End":"01:52.535","Text":"I can easily extract e^2x,"},{"Start":"01:52.535 ","End":"01:55.025","Text":"it\u0027s just t squared minus 1."},{"Start":"01:55.025 ","End":"02:01.140","Text":"What I get is the integral of t squared dt"},{"Start":"02:01.140 ","End":"02:06.835","Text":"over t squared minus 1."},{"Start":"02:06.835 ","End":"02:09.380","Text":"Now, the trick I want to use here"},{"Start":"02:09.380 ","End":"02:16.380","Text":"is to write this as t squared minus 1 plus 1."},{"Start":"02:16.380 ","End":"02:18.210","Text":"What I\u0027m saying is,"},{"Start":"02:18.210 ","End":"02:19.890","Text":"because t squared"},{"Start":"02:19.890 ","End":"02:23.570","Text":"is t squared minus 1 plus 1,"},{"Start":"02:23.570 ","End":"02:26.780","Text":"I can separate this into 2 bits and say,"},{"Start":"02:26.780 ","End":"02:29.090","Text":"first of all, it\u0027s t squared minus 1"},{"Start":"02:29.090 ","End":"02:33.005","Text":"over t squared minus 1 dt"},{"Start":"02:33.005 ","End":"02:35.240","Text":"plus the integral of 1"},{"Start":"02:35.240 ","End":"02:39.850","Text":"over t squared minus 1 dt."},{"Start":"02:40.220 ","End":"02:43.530","Text":"This fraction is just 1,"},{"Start":"02:43.530 ","End":"02:46.500","Text":"so its integral is just t."},{"Start":"02:46.500 ","End":"02:51.715","Text":"The integral of this, it\u0027s 1/2."},{"Start":"02:51.715 ","End":"02:55.790","Text":"The natural logarithm of absolute value"},{"Start":"02:55.790 ","End":"03:02.390","Text":"of t minus 1 over t plus 1 plus a constant."},{"Start":"03:02.390 ","End":"03:05.660","Text":"All that remains now is to substitute back"},{"Start":"03:05.660 ","End":"03:10.549","Text":"instead of t to put this whole expression"},{"Start":"03:10.549 ","End":"03:14.225","Text":"here and it\u0027s going to be a mess."},{"Start":"03:14.225 ","End":"03:18.710","Text":"What we\u0027ll get is the square root of 1"},{"Start":"03:18.710 ","End":"03:27.600","Text":"plus e^2x plus 1/2 natural logarithm of,"},{"Start":"03:27.600 ","End":"03:29.520","Text":"I don\u0027t like this much,"},{"Start":"03:29.520 ","End":"03:31.230","Text":"but I don\u0027t see what I can do,"},{"Start":"03:31.230 ","End":"03:36.080","Text":"square root of 1 plus e^2x minus 1"},{"Start":"03:36.080 ","End":"03:41.450","Text":"over the square root of 1"},{"Start":"03:41.450 ","End":"03:46.820","Text":"plus e^2x plus 1 plus C."},{"Start":"03:46.820 ","End":"03:49.490","Text":"I don\u0027t know if it\u0027s possible to simplify this."},{"Start":"03:49.490 ","End":"03:52.710","Text":"But anyway, this is the answer."}],"ID":6755},{"Watched":false,"Name":"Exercise 18","Duration":"7m 4s","ChapterTopicVideoID":4847,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"Here we have another integral to do by substitution,"},{"Start":"00:04.200 ","End":"00:08.280","Text":"and I\u0027ve pre-written the first steps that we do."},{"Start":"00:08.280 ","End":"00:18.045","Text":"Normally we would try something like t equals the square root of 1 plus 1 over x squared."},{"Start":"00:18.045 ","End":"00:21.120","Text":"But I\u0027ve tried this and it doesn\u0027t work,"},{"Start":"00:21.120 ","End":"00:22.880","Text":"so we have to do something else."},{"Start":"00:22.880 ","End":"00:24.895","Text":"Let me erase this."},{"Start":"00:24.895 ","End":"00:28.925","Text":"What I\u0027m going to do is some Algebraic manipulation."},{"Start":"00:28.925 ","End":"00:33.335","Text":"There\u0027s some fairly standard tricks and I\u0027ll show you what we do here."},{"Start":"00:33.335 ","End":"00:40.485","Text":"I\u0027m going to rewrite the square root of 1 plus 1 over x squared."},{"Start":"00:40.485 ","End":"00:44.330","Text":"First of all, by putting a common denominator for what\u0027s under the square root,"},{"Start":"00:44.330 ","End":"00:46.690","Text":"let\u0027s put it all over x squared."},{"Start":"00:46.690 ","End":"00:52.250","Text":"What I get is x squared plus 1 over x squared."},{"Start":"00:52.250 ","End":"00:57.230","Text":"Now, I take the square root of the numerator and denominator separately."},{"Start":"00:57.230 ","End":"01:02.295","Text":"I get the square root of x squared plus 1 over,"},{"Start":"01:02.295 ","End":"01:08.680","Text":"now normally, the square root of x squared is the absolute value of x."},{"Start":"01:08.680 ","End":"01:10.420","Text":"But in our case,"},{"Start":"01:10.420 ","End":"01:13.420","Text":"we\u0027re given that x is bigger than 0."},{"Start":"01:13.420 ","End":"01:19.070","Text":"For one thing, it makes sure that it\u0027s not equal to 0."},{"Start":"01:21.260 ","End":"01:24.730","Text":"We don\u0027t have a problem with the definition, but it also,"},{"Start":"01:24.730 ","End":"01:27.730","Text":"because it\u0027s bigger than 0, we know if it\u0027s plus or minus."},{"Start":"01:27.730 ","End":"01:31.220","Text":"This in short is just x."},{"Start":"01:31.310 ","End":"01:34.225","Text":"Now we\u0027re going to try the substitution."},{"Start":"01:34.225 ","End":"01:37.315","Text":"We\u0027re going to let t equal just the square root part."},{"Start":"01:37.315 ","End":"01:44.705","Text":"I\u0027m going to substitute t equals the square root of x squared plus 1."},{"Start":"01:44.705 ","End":"01:47.295","Text":"Now where it says, differentiate,"},{"Start":"01:47.295 ","End":"01:49.110","Text":"we do one small step first,"},{"Start":"01:49.110 ","End":"01:50.605","Text":"we don\u0027t want the square root,"},{"Start":"01:50.605 ","End":"01:57.535","Text":"write it as t squared equals x squared plus 1 and now differentiate and we get,"},{"Start":"01:57.535 ","End":"02:05.920","Text":"2tdt is equal to 2xdx."},{"Start":"02:06.800 ","End":"02:12.515","Text":"Of course, I can immediately cancel the 2 on each side."},{"Start":"02:12.515 ","End":"02:16.215","Text":"Now I can extract dx by just saying that,"},{"Start":"02:16.215 ","End":"02:22.960","Text":"dx is equal to tdt over x."},{"Start":"02:23.640 ","End":"02:27.600","Text":"Once again, I want to remind you that x is not 0,"},{"Start":"02:27.600 ","End":"02:30.049","Text":"so this is okay."},{"Start":"02:30.049 ","End":"02:33.475","Text":"Something I should have done earlier would be to"},{"Start":"02:33.475 ","End":"02:36.715","Text":"rewrite this in view of the manipulation here."},{"Start":"02:36.715 ","End":"02:38.275","Text":"What we have now,"},{"Start":"02:38.275 ","End":"02:44.010","Text":"this is the same integral as the square root, is copying from here,"},{"Start":"02:44.010 ","End":"02:47.940","Text":"x squared plus 1 over x,"},{"Start":"02:47.940 ","End":"02:54.370","Text":"dx, and I\u0027m going to replace dx by what it says down here."},{"Start":"02:54.370 ","End":"03:03.440","Text":"We get the integral of the square root of x squared plus 1 over x times"},{"Start":"03:03.440 ","End":"03:05.920","Text":"tdt over x"},{"Start":"03:10.910 ","End":"03:14.700","Text":"and slight rewrite,"},{"Start":"03:14.700 ","End":"03:22.510","Text":"integral of the square root of x squared plus 1 over x squared,"},{"Start":"03:22.510 ","End":"03:26.810","Text":"which is x times x times tdt."},{"Start":"03:27.000 ","End":"03:31.900","Text":"Now this stage we\u0027re supposed to have everything in terms of t. But we"},{"Start":"03:31.900 ","End":"03:37.470","Text":"still have a lot of x\u0027s around, these two places."},{"Start":"03:37.470 ","End":"03:39.450","Text":"What I can do,"},{"Start":"03:39.450 ","End":"03:42.045","Text":"is if I look over here,"},{"Start":"03:42.045 ","End":"03:47.880","Text":"I can replace x squared plus 1 by t"},{"Start":"03:47.880 ","End":"03:53.575","Text":"squared and x squared would just be t squared minus 1."},{"Start":"03:53.575 ","End":"03:55.060","Text":"What we get now,"},{"Start":"03:55.060 ","End":"04:00.235","Text":"is the square root of x squared plus 1 is t squared."},{"Start":"04:00.235 ","End":"04:07.340","Text":"x squared is t squared minus 1 and then tdt."},{"Start":"04:11.200 ","End":"04:16.010","Text":"I should have been writing equals signs there, not too late."},{"Start":"04:16.010 ","End":"04:21.520","Text":"Now, t is also positive because it\u0027s the square root of something."},{"Start":"04:21.520 ","End":"04:25.324","Text":"This comes out to be the integral."},{"Start":"04:25.324 ","End":"04:28.340","Text":"This is t, t with t is t squared,"},{"Start":"04:28.340 ","End":"04:33.240","Text":"t squared over t squared minus 1dt."},{"Start":"04:33.970 ","End":"04:37.345","Text":"This is straightforward enough."},{"Start":"04:37.345 ","End":"04:40.805","Text":"Now we\u0027re going to do another little Algebraic manipulation."},{"Start":"04:40.805 ","End":"04:44.900","Text":"I\u0027m running out of space and I\u0027ll continue up here."},{"Start":"04:44.900 ","End":"04:48.240","Text":"Now I\u0027m going to do something and then I\u0027ll explain."},{"Start":"04:48.240 ","End":"04:55.140","Text":"This expression is equal to 1"},{"Start":"04:55.140 ","End":"05:09.720","Text":"plus 1 over t squared minus 1."},{"Start":"05:10.120 ","End":"05:15.670","Text":"All I did was think of t squared as t squared minus 1 plus 1,"},{"Start":"05:15.670 ","End":"05:20.045","Text":"and the t squared minus 1 over t squared minus 1 is 1."},{"Start":"05:20.045 ","End":"05:22.130","Text":"If you\u0027re not sure about this,"},{"Start":"05:22.130 ","End":"05:23.810","Text":"you can just multiply it out."},{"Start":"05:23.810 ","End":"05:24.980","Text":"Put a common denominator,"},{"Start":"05:24.980 ","End":"05:26.885","Text":"you get t squared minus 1 plus 1."},{"Start":"05:26.885 ","End":"05:29.005","Text":"This is okay."},{"Start":"05:29.005 ","End":"05:32.160","Text":"I now have to do the integral of 1."},{"Start":"05:32.160 ","End":"05:37.970","Text":"The question is, what is the integral of 1 over t squared minus 1?"},{"Start":"05:37.970 ","End":"05:42.755","Text":"I don\u0027t want to waste a lot of time talking about"},{"Start":"05:42.755 ","End":"05:47.180","Text":"the partial fraction solution and you may not have heard of this method,"},{"Start":"05:47.180 ","End":"05:52.440","Text":"but let me just use the integral table and give you the solution for that."},{"Start":"05:53.600 ","End":"05:56.010","Text":"The integral of this,"},{"Start":"05:56.010 ","End":"06:06.675","Text":"is 1/2 natural logarithm of t minus 1 over t plus 1."},{"Start":"06:06.675 ","End":"06:10.975","Text":"As for the rest of it, the integral of 1 is just t,"},{"Start":"06:10.975 ","End":"06:14.015","Text":"and of course, the constant of integration."},{"Start":"06:14.015 ","End":"06:16.460","Text":"But we\u0027re not done yet,"},{"Start":"06:16.460 ","End":"06:21.200","Text":"because we still have to substitute back from t to x."},{"Start":"06:21.200 ","End":"06:28.525","Text":"We get that this is equal to t,"},{"Start":"06:28.525 ","End":"06:33.680","Text":"which is just the square root of x squared plus"},{"Start":"06:33.680 ","End":"06:43.205","Text":"1 plus 1/2 natural logarithm of square root of x squared"},{"Start":"06:43.205 ","End":"06:50.210","Text":"plus 1 minus 1 over square root of x squared plus"},{"Start":"06:50.210 ","End":"06:58.770","Text":"1 plus 1 plus C. It\u0027s not a pretty sight."},{"Start":"06:59.060 ","End":"07:04.800","Text":"I\u0027ll highlight the solution and we are done."}],"ID":4847},{"Watched":false,"Name":"Exercise 19","Duration":"2m 9s","ChapterTopicVideoID":6695,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.430","Text":"In this exercise, we have to compute the integral of e to the power of square root of x,"},{"Start":"00:05.430 ","End":"00:08.625","Text":"and we\u0027re going to try doing it by substitution."},{"Start":"00:08.625 ","End":"00:10.650","Text":"What should we substitute?"},{"Start":"00:10.650 ","End":"00:15.990","Text":"I have an intuition that the square root of x will be good to substitute."},{"Start":"00:15.990 ","End":"00:20.100","Text":"Let that equal t. Before we differentiate,"},{"Start":"00:20.100 ","End":"00:24.000","Text":"let\u0027s square this so we don\u0027t have to deal with square roots,"},{"Start":"00:24.000 ","End":"00:27.035","Text":"so x equals t squared,"},{"Start":"00:27.035 ","End":"00:30.840","Text":"and then dx will equal 2tdt,"},{"Start":"00:31.600 ","End":"00:39.540","Text":"and dx already is extracted but I\u0027ll write it again, dx equals 2tdt."},{"Start":"00:39.540 ","End":"00:42.110","Text":"Just copied it. By luck,"},{"Start":"00:42.110 ","End":"00:44.720","Text":"we already had it in the right form."},{"Start":"00:44.720 ","End":"00:50.090","Text":"What I want to do now is to substitute the square root of x in"},{"Start":"00:50.090 ","End":"00:56.295","Text":"here and I also want to put the dx here."},{"Start":"00:56.295 ","End":"00:59.595","Text":"Let\u0027s see what we end up with from this integral."},{"Start":"00:59.595 ","End":"01:03.860","Text":"Over here, I\u0027ll do the integral of e to the power of"},{"Start":"01:03.860 ","End":"01:09.500","Text":"square root of x is t and dx is 2tdt."},{"Start":"01:11.990 ","End":"01:22.710","Text":"This I can write also as twice the integral of te to the t dt,"},{"Start":"01:22.710 ","End":"01:25.700","Text":"and I happen to be familiar with this integral,"},{"Start":"01:25.700 ","End":"01:27.410","Text":"I\u0027ve done it a few times before,"},{"Start":"01:27.410 ","End":"01:30.035","Text":"and it\u0027s easily done by parts."},{"Start":"01:30.035 ","End":"01:32.525","Text":"If you integrate it by parts,"},{"Start":"01:32.525 ","End":"01:37.520","Text":"what you get is t minus 1 e to"},{"Start":"01:37.520 ","End":"01:43.535","Text":"the power of t. You can check it easily by differentiating a product."},{"Start":"01:43.535 ","End":"01:48.530","Text":"You\u0027ll get 1 times e to the t plus t minus 1 e to the t"},{"Start":"01:48.530 ","End":"01:54.220","Text":"altogether t to the t. The 2 stays here and we add a constant."},{"Start":"01:54.220 ","End":"01:57.305","Text":"Finally, to substitute back instead of t,"},{"Start":"01:57.305 ","End":"01:59.090","Text":"we want the square root of x,"},{"Start":"01:59.090 ","End":"02:03.230","Text":"so the final answer is twice square root of x minus"},{"Start":"02:03.230 ","End":"02:10.180","Text":"1 e to the power of square root of x plus c, and we\u0027re done."}],"ID":6756},{"Watched":false,"Name":"Exercise 20","Duration":"2m 26s","ChapterTopicVideoID":6696,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"Here we have to compute the integral of e to the power of cube root of"},{"Start":"00:04.650 ","End":"00:07.830","Text":"x. I remember not long ago doing e to"},{"Start":"00:07.830 ","End":"00:12.210","Text":"the power of square root of x. Hopefully, it\u0027ll go similarly."},{"Start":"00:12.210 ","End":"00:15.675","Text":"There we used integration by substitution,"},{"Start":"00:15.675 ","End":"00:17.820","Text":"and we substituted the square root of x."},{"Start":"00:17.820 ","End":"00:22.020","Text":"Now we\u0027ll substitute the cube root of x. I\u0027ll see if that works."},{"Start":"00:22.020 ","End":"00:26.040","Text":"Cube root of x well that equal to t. Now,"},{"Start":"00:26.040 ","End":"00:29.185","Text":"before we differentiate will cube both sides,"},{"Start":"00:29.185 ","End":"00:32.880","Text":"so that x is equal to t cubed,"},{"Start":"00:32.880 ","End":"00:42.205","Text":"and then it\u0027s easier to differentiate so we get that dx is equal to 3t squared dt."},{"Start":"00:42.205 ","End":"00:46.670","Text":"Here I just have to copy the last line because it already is extracted."},{"Start":"00:46.670 ","End":"00:52.020","Text":"3t squared dt."},{"Start":"00:52.670 ","End":"00:56.345","Text":"Now, I want to do some substituting."},{"Start":"00:56.345 ","End":"01:00.305","Text":"I want to substitute dx here,"},{"Start":"01:00.305 ","End":"01:07.830","Text":"and I want to substitute the cube root of x here."},{"Start":"01:08.000 ","End":"01:12.265","Text":"What I\u0027ll get from here is,"},{"Start":"01:12.265 ","End":"01:17.825","Text":"the integral of e to the power of t,"},{"Start":"01:17.825 ","End":"01:20.790","Text":"and dx is 3."},{"Start":"01:20.790 ","End":"01:24.990","Text":"I\u0027ll put that here, t squared dt."},{"Start":"01:24.990 ","End":"01:28.570","Text":"This is a well-known integral,"},{"Start":"01:28.570 ","End":"01:33.200","Text":"and this has to be integrated by parts twice."},{"Start":"01:33.200 ","End":"01:36.585","Text":"I\u0027ll just give you the answer."},{"Start":"01:36.585 ","End":"01:42.080","Text":"That this is 3 e to the t times."},{"Start":"01:42.080 ","End":"01:44.150","Text":"You should try this on your own,"},{"Start":"01:44.150 ","End":"01:50.630","Text":"or at least differentiate e to the t times this and see that you get this."},{"Start":"01:50.630 ","End":"01:55.760","Text":"All we have to do now is to substitute back wherever we"},{"Start":"01:55.760 ","End":"02:00.965","Text":"have t we should put cube root of x or x to the power of a third."},{"Start":"02:00.965 ","End":"02:06.429","Text":"What we get is 3 e to the cube root of x,"},{"Start":"02:06.429 ","End":"02:12.745","Text":"and x to the power of a third to the power of 2 is x to the 2/3,"},{"Start":"02:12.745 ","End":"02:14.870","Text":"or I could have written it in another way."},{"Start":"02:14.870 ","End":"02:19.220","Text":"Cube root of x squared minus 2t is the cube root of x,"},{"Start":"02:19.220 ","End":"02:22.760","Text":"but I started writing exponents plus 2,"},{"Start":"02:22.760 ","End":"02:26.820","Text":"plus a constant. That\u0027s the answer."}],"ID":6757},{"Watched":false,"Name":"Exercise 21","Duration":"5m 46s","ChapterTopicVideoID":6697,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"In this exercise, we have to compute"},{"Start":"00:01.980 ","End":"00:03.195","Text":"the following integral,"},{"Start":"00:03.195 ","End":"00:06.750","Text":"1 over square root of x plus cube root of x."},{"Start":"00:06.750 ","End":"00:08.550","Text":"You don\u0027t often see there\u0027s a mixture"},{"Start":"00:08.550 ","End":"00:09.990","Text":"of square root and cube root."},{"Start":"00:09.990 ","End":"00:11.490","Text":"Anyway, we\u0027ll try to do this"},{"Start":"00:11.490 ","End":"00:14.010","Text":"by the technique of integration"},{"Start":"00:14.010 ","End":"00:15.525","Text":"by substitution."},{"Start":"00:15.525 ","End":"00:17.190","Text":"What are we going to substitute,"},{"Start":"00:17.190 ","End":"00:19.275","Text":"the square root or the cube root?"},{"Start":"00:19.275 ","End":"00:21.660","Text":"Well, turns out that in these cases,"},{"Start":"00:21.660 ","End":"00:23.775","Text":"you find common denominator."},{"Start":"00:23.775 ","End":"00:26.760","Text":"Say this is to the power of 1/2"},{"Start":"00:26.760 ","End":"00:28.740","Text":"and this is to the power of 1/3,"},{"Start":"00:28.740 ","End":"00:31.320","Text":"then 1/6 is a common denominator."},{"Start":"00:31.320 ","End":"00:34.710","Text":"We take the 6th root of x"},{"Start":"00:34.710 ","End":"00:38.100","Text":"to substitute and let that equal to t."},{"Start":"00:38.100 ","End":"00:40.080","Text":"Before we differentiate,"},{"Start":"00:40.080 ","End":"00:43.350","Text":"we take it as x equals t^6"},{"Start":"00:43.350 ","End":"00:45.525","Text":"by raising to the 6th power."},{"Start":"00:45.525 ","End":"00:52.270","Text":"Now, dx is equal to 6t^5th dt."},{"Start":"00:52.270 ","End":"00:54.770","Text":"There\u0027s no need to extract anything,"},{"Start":"00:54.770 ","End":"00:56.630","Text":"but just to fill in this place,"},{"Start":"00:56.630 ","End":"00:59.905","Text":"I\u0027ll just copy the above line exactly as is."},{"Start":"00:59.905 ","End":"01:03.260","Text":"Now, I want to make some replacements."},{"Start":"01:03.260 ","End":"01:04.700","Text":"I want to take dx,"},{"Start":"01:04.700 ","End":"01:07.460","Text":"and put it here as to what it equals."},{"Start":"01:07.460 ","End":"01:10.820","Text":"I want to take the 6th root of x"},{"Start":"01:10.820 ","End":"01:13.100","Text":"and substitute it here."},{"Start":"01:13.100 ","End":"01:15.170","Text":"But I don\u0027t exactly have 6th root of x."},{"Start":"01:15.170 ","End":"01:18.740","Text":"I\u0027m going to substitute in here and in here,"},{"Start":"01:18.740 ","End":"01:22.380","Text":"but not exactly so I\u0027ll just highlight below."},{"Start":"01:22.380 ","End":"01:25.265","Text":"I\u0027ll show you what I mean in a moment."},{"Start":"01:25.265 ","End":"01:26.570","Text":"Continuing over here,"},{"Start":"01:26.570 ","End":"01:30.295","Text":"I write this integral of 1 over."},{"Start":"01:30.295 ","End":"01:32.430","Text":"Now, the square root of x"},{"Start":"01:32.430 ","End":"01:35.850","Text":"is like the 6th root of x to the power of 3,"},{"Start":"01:35.850 ","End":"01:37.260","Text":"because this is to the power of 1/2"},{"Start":"01:37.260 ","End":"01:39.625","Text":"and this is to the power of a 6th."},{"Start":"01:39.625 ","End":"01:42.380","Text":"Anyway, it comes out that this is this cubed,"},{"Start":"01:42.380 ","End":"01:44.690","Text":"so this is t cubed."},{"Start":"01:44.690 ","End":"01:47.390","Text":"Likewise, cube root of x is"},{"Start":"01:47.390 ","End":"01:48.870","Text":"the 6th root of x squared."},{"Start":"01:48.870 ","End":"01:51.125","Text":"This comes out to be t squared."},{"Start":"01:51.125 ","End":"01:53.450","Text":"Now, dx is 60 to the 5th dt,"},{"Start":"01:53.450 ","End":"01:57.600","Text":"so I write it."},{"Start":"01:57.600 ","End":"02:00.480","Text":"The 6 here will need t^5th,"},{"Start":"02:00.480 ","End":"02:06.590","Text":"so I will change this into a t^5th dt."},{"Start":"02:06.590 ","End":"02:07.820","Text":"We could divide top"},{"Start":"02:07.820 ","End":"02:12.130","Text":"and bottom by t squared."},{"Start":"02:12.130 ","End":"02:15.560","Text":"We get 6 times the integral"},{"Start":"02:15.560 ","End":"02:22.550","Text":"of t cubed over t plus 1 dt."},{"Start":"02:22.550 ","End":"02:25.580","Text":"It\u0027s possible to do this with another substitution,"},{"Start":"02:25.580 ","End":"02:27.260","Text":"say, s equals t plus 1,"},{"Start":"02:27.260 ","End":"02:30.395","Text":"but I tried it and it\u0027s quite long."},{"Start":"02:30.395 ","End":"02:32.450","Text":"I\u0027m going to do it by another method."},{"Start":"02:32.450 ","End":"02:35.330","Text":"I\u0027ll do it by long division of polynomials."},{"Start":"02:35.330 ","End":"02:36.765","Text":"I hope you\u0027ve learned this."},{"Start":"02:36.765 ","End":"02:39.450","Text":"But if not, you can easily follow this,"},{"Start":"02:39.450 ","End":"02:40.840","Text":"I hope."},{"Start":"02:40.840 ","End":"02:42.920","Text":"Here, we have t cubed,"},{"Start":"02:42.920 ","End":"02:44.960","Text":"that\u0027s the 1 from the top."},{"Start":"02:44.960 ","End":"02:46.190","Text":"Sometimes you write"},{"Start":"02:46.190 ","End":"02:49.890","Text":"plus 0t squared plus 0t plus 0,"},{"Start":"02:49.890 ","End":"02:51.180","Text":"but you don\u0027t have to,"},{"Start":"02:51.180 ","End":"02:54.100","Text":"and then dividing this by t plus 1."},{"Start":"02:54.100 ","End":"02:56.540","Text":"Now, you only look at the highest powers,"},{"Start":"02:56.540 ","End":"02:59.750","Text":"t into t cubed goes t squared times."},{"Start":"02:59.750 ","End":"03:01.820","Text":"T squared times t plus 1"},{"Start":"03:01.820 ","End":"03:07.140","Text":"is t cubed plus t squared,"},{"Start":"03:07.140 ","End":"03:08.610","Text":"and now subtract."},{"Start":"03:08.610 ","End":"03:10.550","Text":"This minus this is nothing,"},{"Start":"03:10.550 ","End":"03:12.530","Text":"and 0 minus t squared"},{"Start":"03:12.530 ","End":"03:14.810","Text":"is minus t squared."},{"Start":"03:14.810 ","End":"03:19.395","Text":"Drop the next term down, plus 0t."},{"Start":"03:19.395 ","End":"03:25.550","Text":"T into minus t squared goes minus t times."},{"Start":"03:25.550 ","End":"03:27.320","Text":"Multiplying, we get minus"},{"Start":"03:27.320 ","End":"03:29.690","Text":"t squared minus t."},{"Start":"03:29.690 ","End":"03:34.590","Text":"Again, subtract and this cancels,"},{"Start":"03:34.590 ","End":"03:38.670","Text":"and we get plus t plus 0."},{"Start":"03:38.670 ","End":"03:44.985","Text":"T plus 1, t goes into t 1 time, t plus 1."},{"Start":"03:44.985 ","End":"03:47.550","Text":"Now, subtract minus 1."},{"Start":"03:47.550 ","End":"03:51.800","Text":"This is the quotient"},{"Start":"03:51.800 ","End":"03:52.925","Text":"and this is the remainder."},{"Start":"03:52.925 ","End":"03:57.920","Text":"This thing equals 6 times the integral."},{"Start":"03:57.920 ","End":"04:00.860","Text":"Now, what we get is the quotient,"},{"Start":"04:00.860 ","End":"04:04.650","Text":"t squared minus t plus 1."},{"Start":"04:04.650 ","End":"04:06.960","Text":"The remainder minus 1"},{"Start":"04:06.960 ","End":"04:10.410","Text":"also has to go over t plus 1 dt,"},{"Start":"04:10.410 ","End":"04:15.015","Text":"and this equals 6 times."},{"Start":"04:15.015 ","End":"04:16.910","Text":"Now, the integral of this is"},{"Start":"04:16.910 ","End":"04:23.555","Text":"t cubed over 3 minus t squared over 2"},{"Start":"04:23.555 ","End":"04:33.320","Text":"plus t minus natural log of t plus 1."},{"Start":"04:33.320 ","End":"04:35.410","Text":"I have to divide this by 1,"},{"Start":"04:35.410 ","End":"04:37.000","Text":"because that\u0027s the coefficient of t,"},{"Start":"04:37.000 ","End":"04:39.280","Text":"but dividing by 1 doesn\u0027t change it,"},{"Start":"04:39.280 ","End":"04:42.605","Text":"plus the constant."},{"Start":"04:42.605 ","End":"04:46.060","Text":"Finally, what we have to do"},{"Start":"04:46.060 ","End":"04:49.150","Text":"is just write it with a reverse"},{"Start":"04:49.150 ","End":"04:51.370","Text":"substitution t in terms of x."},{"Start":"04:51.370 ","End":"04:55.040","Text":"I\u0027m letting t, it\u0027s also equal"},{"Start":"04:55.040 ","End":"04:57.620","Text":"to x to the power of 1/6th."},{"Start":"04:57.620 ","End":"05:00.765","Text":"I\u0027ll use whatever form."},{"Start":"05:00.765 ","End":"05:06.969","Text":"What I have is 6 times t cubed is,"},{"Start":"05:06.969 ","End":"05:08.170","Text":"you know what?"},{"Start":"05:08.170 ","End":"05:09.505","Text":"I\u0027ll leave it in terms of roots."},{"Start":"05:09.505 ","End":"05:12.630","Text":"T cubed is x^6th,"},{"Start":"05:12.630 ","End":"05:15.860","Text":"the power of 3 is x^1/2 is square root of x."},{"Start":"05:15.860 ","End":"05:20.700","Text":"Square root of x over 3."},{"Start":"05:20.700 ","End":"05:23.185","Text":"T squared would be x^1/3,"},{"Start":"05:23.185 ","End":"05:25.205","Text":"which is the cube root of x,"},{"Start":"05:25.205 ","End":"05:29.025","Text":"cube root of x over 2,"},{"Start":"05:29.025 ","End":"05:32.540","Text":"plus the 6th root of x"},{"Start":"05:32.540 ","End":"05:39.230","Text":"minus natural log of 6th root of x"},{"Start":"05:39.230 ","End":"05:43.490","Text":"plus 1 plus a constant."},{"Start":"05:43.490 ","End":"05:46.830","Text":"We are done."}],"ID":6758},{"Watched":false,"Name":"Exercise 22","Duration":"4m 59s","ChapterTopicVideoID":6698,"CourseChapterTopicPlaylistID":2676,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.440","Text":"In this exercise, we have to compute the integral of arctangent of square root of x."},{"Start":"00:07.440 ","End":"00:12.300","Text":"We\u0027ll use the technique of integration by substitution."},{"Start":"00:12.300 ","End":"00:14.340","Text":"The question is what to substitute?"},{"Start":"00:14.340 ","End":"00:18.600","Text":"I think we should go with the square root of x and let"},{"Start":"00:18.600 ","End":"00:23.655","Text":"that equal t. Before we differentiate, let\u0027s square it."},{"Start":"00:23.655 ","End":"00:25.260","Text":"That way it will be easier,"},{"Start":"00:25.260 ","End":"00:31.660","Text":"x is t squared and then dx is 2t, dt."},{"Start":"00:31.660 ","End":"00:35.630","Text":"We don\u0027t have to extract dx because already is extracted,"},{"Start":"00:35.630 ","End":"00:39.185","Text":"so I\u0027ll just copy it just to fill this space."},{"Start":"00:39.185 ","End":"00:45.860","Text":"Now what I want to do is to make a few substitutions in a different sense."},{"Start":"00:45.860 ","End":"00:52.550","Text":"What I\u0027m going to do is take the square root of x and put it here"},{"Start":"00:52.550 ","End":"01:00.315","Text":"and I\u0027m also going to substitute the dx here."},{"Start":"01:00.315 ","End":"01:04.340","Text":"What I get from this integral is"},{"Start":"01:04.340 ","End":"01:13.995","Text":"the integral arctangent of t and dx is 2t, dt."},{"Start":"01:13.995 ","End":"01:17.595","Text":"Now the question is, how am I going to do this integral?"},{"Start":"01:17.595 ","End":"01:20.780","Text":"Well, I\u0027ll do it quickly,"},{"Start":"01:20.780 ","End":"01:25.324","Text":"but it\u0027s going to be by parts assuming that you know how to integrate by parts."},{"Start":"01:25.324 ","End":"01:28.520","Text":"Remember that we need the integral of"},{"Start":"01:28.520 ","End":"01:36.555","Text":"udv is uv minus the integral of vdu."},{"Start":"01:36.555 ","End":"01:40.850","Text":"The question is which of these 2 is going to be u and"},{"Start":"01:40.850 ","End":"01:44.510","Text":"which is going to be v. My experience shows"},{"Start":"01:44.510 ","End":"01:53.160","Text":"that this 1 is going to be u and this bit here is going to be dv."},{"Start":"01:53.160 ","End":"02:01.210","Text":"What we get is du is equal to the derivative of arctangent"},{"Start":"02:01.210 ","End":"02:09.745","Text":"of t is 1 over 1 plus t squared."},{"Start":"02:09.745 ","End":"02:15.730","Text":"It\u0027s 1 over 1 plus t squared dt"},{"Start":"02:15.730 ","End":"02:25.610","Text":"and if dv is 2t,"},{"Start":"02:25.610 ","End":"02:29.660","Text":"then this is exactly the derivative of t squared."},{"Start":"02:29.660 ","End":"02:35.570","Text":"We can see that v will equal t squared and you can do it the other way around."},{"Start":"02:35.570 ","End":"02:38.240","Text":"Then dv will be 2t dt."},{"Start":"02:38.240 ","End":"02:41.215","Text":"What we have to get is uv."},{"Start":"02:41.215 ","End":"02:43.490","Text":"This integral is uv,"},{"Start":"02:43.490 ","End":"02:46.730","Text":"which is arctangent t times t squared."},{"Start":"02:46.730 ","End":"02:52.715","Text":"I\u0027ll write it as t squared arctangent of t,"},{"Start":"02:52.715 ","End":"02:59.450","Text":"the uv minus the integral of vdu."},{"Start":"02:59.450 ","End":"03:06.280","Text":"v is t squared and du is 1 over t squared dt."},{"Start":"03:07.490 ","End":"03:17.430","Text":"This we do by the usual trick of saying that t squared is 1 plus t squared minus 1."},{"Start":"03:17.810 ","End":"03:20.790","Text":"We get from here,"},{"Start":"03:20.790 ","End":"03:24.435","Text":"this is just the scratch area."},{"Start":"03:24.435 ","End":"03:33.605","Text":"From here we get t squared arctangent of t minus,"},{"Start":"03:33.605 ","End":"03:36.380","Text":"if I\u0027m letting it be 1 plus t squared minus 1,"},{"Start":"03:36.380 ","End":"03:40.760","Text":"the 1 plus t squared over 1 plus t squared cancels."},{"Start":"03:40.760 ","End":"03:45.740","Text":"What I get in here basically is 1 and then"},{"Start":"03:45.740 ","End":"03:52.100","Text":"minus 1 over 1 plus t squared dt,"},{"Start":"03:52.100 ","End":"03:59.585","Text":"which is equal to again t squared arctangent t minus"},{"Start":"03:59.585 ","End":"04:06.870","Text":"integral of 1 is t and minus a minus is plus."},{"Start":"04:06.870 ","End":"04:11.010","Text":"The integral of 1 over 1 plus t squared is the arctangent."},{"Start":"04:11.010 ","End":"04:12.705","Text":"We just did this here,"},{"Start":"04:12.705 ","End":"04:20.170","Text":"plus arctangent t plus a constant."},{"Start":"04:21.200 ","End":"04:25.820","Text":"All we have to do now is back substitute."},{"Start":"04:25.820 ","End":"04:27.560","Text":"Meaning wherever we see t,"},{"Start":"04:27.560 ","End":"04:29.495","Text":"we put square root of x."},{"Start":"04:29.495 ","End":"04:33.610","Text":"We have square root of x squared is x"},{"Start":"04:33.610 ","End":"04:38.755","Text":"and we also have another arctangent t. Let\u0027s gather these 2 together."},{"Start":"04:38.755 ","End":"04:40.525","Text":"It\u0027s t squared plus 1,"},{"Start":"04:40.525 ","End":"04:49.940","Text":"which is x plus 1 arctangent of t. But we don\u0027t put t. I put square root of x minus t,"},{"Start":"04:49.940 ","End":"04:53.210","Text":"which is the square root of x plus"},{"Start":"04:53.210 ","End":"05:00.120","Text":"c. This is the answer unless I made a mistake and we\u0027re done."}],"ID":6759}],"Thumbnail":null,"ID":2676}]

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