WEBVTT 00:00:01.010 --> 00:00:03.854 Sometimes and apparently complicated integral can 00:00:03.854 --> 00:00:08.120 be evaluated by first of all making a substitution. 00:00:09.140 --> 00:00:13.573 The effect of this is to change the variable, say from X to 00:00:13.573 --> 00:00:14.596 another variable U. 00:00:15.350 --> 00:00:18.150 It changes the integrand. That's the quantity that 00:00:18.150 --> 00:00:19.550 we're trying to integrate. 00:00:20.620 --> 00:00:23.076 And if we're dealing with definite integrals, those 00:00:23.076 --> 00:00:25.532 with limits than the limits may change too. 00:00:26.730 --> 00:00:29.824 Before we look at our first example, I want to give you a 00:00:29.824 --> 00:00:32.442 preliminary result which will be very useful in all of the 00:00:32.442 --> 00:00:33.632 examples that we look at. 00:00:34.570 --> 00:00:36.790 Suppose we have a variable U. 00:00:37.860 --> 00:00:42.852 Which is a function of X, so you is a function of X. 00:00:43.860 --> 00:00:46.060 Suppose we can differentiate this 00:00:46.060 --> 00:00:49.140 function of X and workout du DX. 00:00:53.590 --> 00:00:56.090 Then a quantity called the 00:00:56.090 --> 00:00:59.590 differential du is given by du DX. 00:01:02.780 --> 00:01:07.114 Multiply by DX and this will be a particularly important result 00:01:07.114 --> 00:01:12.630 in all of the examples that we do. So, for example, if I had 00:01:12.630 --> 00:01:17.358 you equals 1 - 2 X, so there's my function of X. 00:01:18.280 --> 00:01:21.040 We can differentiate this du DX. 00:01:24.660 --> 00:01:27.810 Well, in this case du DX will be minus 2. 00:01:30.350 --> 00:01:34.715 Then the differential du is given by du DX. 00:01:35.350 --> 00:01:39.360 Minus 2 multiplied by DX. 00:01:41.800 --> 00:01:46.300 So this formula will be very useful in all the examples that 00:01:46.300 --> 00:01:51.175 we do. Let's have a look at our first example of integration by 00:01:51.175 --> 00:01:54.175 substitution. Suppose we want to perform this integration. 00:01:54.175 --> 00:01:57.925 Suppose we want to find the integral of X +4. 00:02:01.120 --> 00:02:05.878 All raised to the power 5 and we want to integrate this with 00:02:05.878 --> 00:02:06.976 respect to X. 00:02:10.100 --> 00:02:13.532 Now the problem here in this example is the X +4. 00:02:14.270 --> 00:02:17.240 This is the complicated bit and the reason why it's 00:02:17.240 --> 00:02:20.210 complicated is because if this was just a single variable, 00:02:20.210 --> 00:02:24.071 say X would be integrating X to the power 5, and we all 00:02:24.071 --> 00:02:27.635 know how to do that already. So this is the problem. We'd 00:02:27.635 --> 00:02:31.199 like this to be a single variable, and so we make a 00:02:31.199 --> 00:02:33.872 substitution and let you be this quantity X +4. 00:02:39.080 --> 00:02:42.995 Let's see what happens when we make this substitution. 00:02:44.110 --> 00:02:49.076 This integral is going to become the integral X +4 is going to 00:02:49.076 --> 00:02:50.604 become just simply you. 00:02:52.510 --> 00:02:54.365 We've got you to the power 5. 00:02:55.880 --> 00:03:00.374 And then we have to be a little bit careful because we've got to 00:03:00.374 --> 00:03:04.226 take care of this quantity an appropriate way as well. Now we 00:03:04.226 --> 00:03:08.078 do that using the results I've just given you. We workout the 00:03:08.078 --> 00:03:10.967 differential du. Now remember that du is du DX. 00:03:12.700 --> 00:03:14.338 Multiplied by DX. 00:03:16.550 --> 00:03:20.390 In this case, du DX is just one. If you equals X 00:03:20.390 --> 00:03:23.910 +4 du DX, the derivative of you will be just one. 00:03:25.960 --> 00:03:29.652 So in this first example with a nice simple case in which du 00:03:29.652 --> 00:03:33.912 is just equal to 1, DX or du is equal to DX. So when we 00:03:33.912 --> 00:03:36.468 make the substitution of, the DX can become simply. 00:03:38.030 --> 00:03:38.450 See you. 00:03:39.640 --> 00:03:43.012 Now this is a much simpler integral than the one we started 00:03:43.012 --> 00:03:47.227 with and will be able to finish it off in the usual way just by 00:03:47.227 --> 00:03:50.880 increasing the power by one, so will get you to the Palace 6. 00:03:51.760 --> 00:03:53.608 We divide by the new power. 00:03:54.300 --> 00:03:56.720 And we don't forget to add the constant of integration. 00:03:59.100 --> 00:04:01.908 Now, for all intents and purposes, that's the problem 00:04:01.908 --> 00:04:05.028 solved. However, it's normal to go back to our original 00:04:05.028 --> 00:04:07.212 variables. Remember, we introduced this new variable 00:04:07.212 --> 00:04:10.644 you, so we go back to the variable of the original 00:04:10.644 --> 00:04:12.204 problem, which was X. Remember 00:04:12.204 --> 00:04:14.970 that you. Is equal to X +4. 00:04:16.330 --> 00:04:18.760 So replacing you by X +4. 00:04:21.080 --> 00:04:26.428 We have X +4 to the power six or divided by 6 plus a 00:04:26.428 --> 00:04:27.574 constant of integration. 00:04:29.340 --> 00:04:33.309 And that's our first example of integration by substitution. 00:04:37.110 --> 00:04:39.740 Let's have a look at another example. Suppose this time 00:04:39.740 --> 00:04:42.107 that we want to integrate a trigonometric function and 00:04:42.107 --> 00:04:44.737 the one I'm going to look at is the cosine. 00:04:46.240 --> 00:04:48.048 Of three X +4. 00:04:50.800 --> 00:04:52.276 And we want to integrate this. 00:04:53.820 --> 00:04:54.968 With respect to X. 00:04:55.930 --> 00:04:59.450 I'm going to assume that we already know how to integrate 00:04:59.450 --> 00:05:03.290 cosine X Sign X, the standard trig functions. This is a bit 00:05:03.290 --> 00:05:06.810 more complicated, and the reason it's more complicated is the 3X 00:05:06.810 --> 00:05:08.090 plus four in here. 00:05:09.030 --> 00:05:13.430 So as before, we make a substitution to change this 00:05:13.430 --> 00:05:18.270 into just a single variable, so I'm going to write you 00:05:18.270 --> 00:05:20.030 equals 3X plus 4. 00:05:22.230 --> 00:05:26.000 The effect of doing that is to change the integral to this one. 00:05:26.000 --> 00:05:27.450 The integral of the cosine. 00:05:28.980 --> 00:05:32.130 And instead of 3X plus four, we're now going to 00:05:32.130 --> 00:05:34.965 have just simply. You are much simpler problem than 00:05:34.965 --> 00:05:36.540 the one we started with. 00:05:38.220 --> 00:05:41.542 As before, we need to take particular care with this term. 00:05:41.542 --> 00:05:46.380 Here the DX. And we do that with our standard results that du. 00:05:48.710 --> 00:05:50.680 Is equal to du DX. 00:05:51.490 --> 00:05:52.180 DX 00:05:53.870 --> 00:05:57.379 now what's du DX? In this example? Well, you equals 3X 00:05:57.379 --> 00:06:00.250 plus four, and if we differentiate this function with 00:06:00.250 --> 00:06:03.759 respect to X, will get that du DX is just three. 00:06:05.000 --> 00:06:10.656 So our differential du is simply 3D XDU is 3D X if we just 00:06:10.656 --> 00:06:15.504 rearrange this will be able to replace the DX by something in 00:06:15.504 --> 00:06:21.564 terms of du over here. So if du is 3 DX then dividing both sides 00:06:21.564 --> 00:06:25.604 by three, we can write down that DX is 1/3. 00:06:27.170 --> 00:06:27.960 Of du 00:06:29.660 --> 00:06:34.524 so the DX in here is replaced by 1/3 of du. Now I'll put the du 00:06:34.524 --> 00:06:38.476 there and the factor of 1/3 can be put outside the integral sign 00:06:38.476 --> 00:06:41.820 like that. A constant factor can immediately be moved outside the 00:06:41.820 --> 00:06:45.906 integral. Now this is very straightforward to finish. We 00:06:45.906 --> 00:06:50.370 all know that the integral of the cosine of you with respect 00:06:50.370 --> 00:06:52.230 to you is sign you. 00:06:54.040 --> 00:06:55.440 Plus a constant of integration. 00:06:57.620 --> 00:07:02.317 And as before, we revert to our original variables using the 00:07:02.317 --> 00:07:07.014 given substitution. You was 3X plus four, so this becomes 1/3 00:07:07.014 --> 00:07:10.003 of the sign of three X +4. 00:07:11.630 --> 00:07:13.230 Plus a constant of integration. 00:07:15.220 --> 00:07:19.170 And that is our second example of integration by substitution. 00:07:20.630 --> 00:07:24.582 Now before I go on, I want to generalize this a little bit. 00:07:24.582 --> 00:07:27.926 This particular example we had the cosine of a constant times 00:07:27.926 --> 00:07:30.966 X plus another constant and let's just generalize that so 00:07:30.966 --> 00:07:34.006 we can deal with any situations where we have a 00:07:34.006 --> 00:07:37.350 cosine of a constant times X plus a constant. So let's 00:07:37.350 --> 00:07:39.174 suppose we look at this example. 00:07:41.140 --> 00:07:45.716 Suppose we want to integrate the cosine of AX plus B. 00:07:47.690 --> 00:07:49.158 With respect to X. 00:07:50.160 --> 00:07:52.868 When A&B are constants. 00:07:59.930 --> 00:08:03.188 Again, the problem is the quantity in the brackets 00:08:03.188 --> 00:08:06.446 here, and we avoid the problem by making a 00:08:06.446 --> 00:08:10.428 substitution. We let you be all this quantity X plus B. 00:08:14.140 --> 00:08:19.780 That changes the integral to the integral of the cosine and X 00:08:19.780 --> 00:08:22.130 Plus B becomes simply you. 00:08:24.540 --> 00:08:28.193 We've got to deal with the DX, and we do that by calculating 00:08:28.193 --> 00:08:29.036 the Differentials Du. 00:08:30.250 --> 00:08:34.270 Du will be du DX, which in this case is simply A. 00:08:35.860 --> 00:08:36.590 DX 00:08:38.730 --> 00:08:42.902 and this result will allow us to replace the DX in here or DX 00:08:42.902 --> 00:08:44.690 will be one over a du. 00:08:50.570 --> 00:08:55.512 So the DX here will become one over a DUI, right? The du there 00:08:55.512 --> 00:08:59.042 under one over a being a constant factor. I'll bring 00:08:59.042 --> 00:09:00.101 straight outside here. 00:09:02.770 --> 00:09:05.666 This is straightforward to finish off because the 00:09:05.666 --> 00:09:10.372 integral of the cosine you is just the sign of you plus a 00:09:10.372 --> 00:09:11.458 constant of integration. 00:09:14.310 --> 00:09:17.970 In terms of our original variables, will have sign and 00:09:17.970 --> 00:09:20.166 you is a X plus B. 00:09:24.340 --> 00:09:25.770 Plus a constant of integration. 00:09:27.810 --> 00:09:31.038 And that's a general result. We can use any time that we 00:09:31.038 --> 00:09:33.997 have to integrate the cosine of a quantity like this. This 00:09:33.997 --> 00:09:37.225 is an example of a linear function X plus be, so we've 00:09:37.225 --> 00:09:40.184 got the cosine of a linear linear function, and we can 00:09:40.184 --> 00:09:43.143 use this results anytime we want to integrate it. So for 00:09:43.143 --> 00:09:45.833 example, if I write down, what's the integral of the 00:09:45.833 --> 00:09:47.178 cosine of Seven X +3? 00:09:49.460 --> 00:09:52.583 Then immediately we recognize that the A is 7. 00:09:53.830 --> 00:09:57.964 The B is 3 and we use this general result to state that 00:09:57.964 --> 00:09:59.872 this integral is one over A. 00:10:00.110 --> 00:10:01.538 Which will be one over 7. 00:10:02.980 --> 00:10:04.080 The sign. 00:10:05.200 --> 00:10:08.428 Of the original content in the brackets, which was Seven X +3. 00:10:10.720 --> 00:10:13.690 Plus a constant of integration. So anytime we get 00:10:13.690 --> 00:10:17.320 a function to integrate, which is the cosine of a linear 00:10:17.320 --> 00:10:19.300 function, we can use this result. 00:10:20.690 --> 00:10:22.898 There's a very similar results for integrating sign, 00:10:22.898 --> 00:10:26.486 and I won't prove it. I'll leave it for you to to prove 00:10:26.486 --> 00:10:29.522 for yourself, but the result is this one that the integral 00:10:29.522 --> 00:10:30.350 of the sign. 00:10:31.900 --> 00:10:32.920 AX plus B. 00:10:35.990 --> 00:10:39.630 With respect to X will be minus one over A. 00:10:40.290 --> 00:10:43.360 Cosine of X plus B. 00:10:44.680 --> 00:10:46.556 Plus a constant of integration, and that's 00:10:46.556 --> 00:10:49.236 another standard result you should be aware of, and you 00:10:49.236 --> 00:10:51.380 should now be able to prove for yourself. 00:10:52.920 --> 00:10:53.450 OK. 00:10:58.040 --> 00:10:59.324 Let's have a look at another 00:10:59.324 --> 00:11:06.336 example. Suppose we want to find the integral of 1 / 1 - 2 X. 00:11:07.040 --> 00:11:08.248 With respect to X. 00:11:10.330 --> 00:11:13.750 Now I'm going to assume that if you had a single letter 00:11:13.750 --> 00:11:17.170 down here, if they've just been an X down here, so we 00:11:17.170 --> 00:11:20.590 we're integrating one over X, you'd know how to do that just 00:11:20.590 --> 00:11:24.295 by the. By that the integral of one over X with respect to 00:11:24.295 --> 00:11:28.000 X is the natural logarithm of the modulus of X. So I'm going 00:11:28.000 --> 00:11:30.850 to assume that you know how to do that already. 00:11:32.440 --> 00:11:36.376 So if we can convert this integral here into one where we 00:11:36.376 --> 00:11:38.344 just got one over a single 00:11:38.344 --> 00:11:43.080 variable. Perhaps we will have to proceed, so will make a 00:11:43.080 --> 00:11:45.976 substitution, and the substitution will make is U 00:11:45.976 --> 00:11:47.786 equals 1 - 2 X. 00:11:49.830 --> 00:11:53.650 What will that do to the integral? Well, the integral 00:11:53.650 --> 00:11:56.706 will become the integral of 1 divided by. 00:11:58.020 --> 00:12:03.870 1 - 2 X will become you much simpler. We have to take care of 00:12:03.870 --> 00:12:06.210 the DX in an appropriate way. 00:12:07.430 --> 00:12:12.536 As before DU is du DX. 00:12:14.380 --> 00:12:15.868 Multiply by DX. 00:12:17.650 --> 00:12:19.726 What's du DX in this case? 00:12:22.150 --> 00:12:24.047 Well, you is 1 - 2 X. 00:12:24.630 --> 00:12:28.998 So do you DX the derivative of this quantity is just minus 2. 00:12:31.610 --> 00:12:32.320 DX 00:12:33.820 --> 00:12:38.560 so when we want to replace the DX in the substitution process, 00:12:38.560 --> 00:12:40.930 we can replace DX by du. 00:12:43.230 --> 00:12:45.980 Divided by minus two, which is minus 1/2 of DS. 00:12:47.560 --> 00:12:52.735 So the DX here becomes du and the factor of a minus 1/2. I can 00:12:52.735 --> 00:12:55.840 right outside being a constant factor. Now this is 00:12:55.840 --> 00:12:59.290 straightforward to finish off because the integral of one over 00:12:59.290 --> 00:13:01.705 you do. You will be just the 00:13:01.705 --> 00:13:03.860 logarithm. Of the modulus of you. 00:13:05.820 --> 00:13:11.507 So we've minus half natural logarithm of the modulus of you. 00:13:12.700 --> 00:13:14.190 Plus a constant of integration. 00:13:16.690 --> 00:13:20.842 Nearly finished or we do is go back to our original variables. 00:13:20.842 --> 00:13:25.340 Now. Remember that the original variable was U equals 1 - 2 X, 00:13:25.340 --> 00:13:29.146 so this will become minus 1/2 natural logarithm of the modulus 00:13:29.146 --> 00:13:30.876 of 1 - 2 X. 00:13:31.620 --> 00:13:33.150 Plus a constant of integration. 00:13:34.610 --> 00:13:35.678 And that's another example. 00:13:37.610 --> 00:13:40.718 It's very easy to generalize this to any cases of the form 00:13:40.718 --> 00:13:44.085 where you have one divided by a linear function of X. Let's just 00:13:44.085 --> 00:13:47.711 see how we can do that, and then we'll get a very useful general 00:13:47.711 --> 00:13:53.820 result. Suppose we want to integrate 1 divided by a X 00:13:53.820 --> 00:13:54.790 plus B. 00:13:56.150 --> 00:13:58.158 We want to integrate with respect to X. 00:13:59.180 --> 00:14:04.240 The substitution we make is U equals a X plus B. 00:14:06.990 --> 00:14:07.790 Do you? 00:14:09.330 --> 00:14:13.280 Will be du DX, which in this case differentiating this 00:14:13.280 --> 00:14:18.020 function will just leave us the constant a do you will be 00:14:18.020 --> 00:14:23.550 equal to a times DX. So when we want to replace the DX we 00:14:23.550 --> 00:14:27.895 will do so by replacing it with one over a du. 00:14:29.360 --> 00:14:33.960 What will that do to our integral? Well, the integral 00:14:33.960 --> 00:14:37.640 will become the integral of 1 divided by. 00:14:39.050 --> 00:14:41.700 AX plus B becomes you. 00:14:43.840 --> 00:14:48.169 And the DX becomes one over a du. I'll write the du here 00:14:48.169 --> 00:14:51.832 and the one over a being a constant factor. Uh, bring 00:14:51.832 --> 00:14:52.498 straight outside. 00:14:55.870 --> 00:14:58.710 Now this is very familiar, straightforward for us to finish 00:14:58.710 --> 00:15:00.698 this the integral of one over U. 00:15:02.500 --> 00:15:07.336 Will be the natural logarithm of the modulus of you plus a 00:15:07.336 --> 00:15:08.545 constant of integration. 00:15:10.800 --> 00:15:15.280 Nearly finished, we just return to our original variables, one 00:15:15.280 --> 00:15:20.656 over a natural logarithm of the modulus U was X plus B. 00:15:24.480 --> 00:15:25.870 Plus a constant of integration. 00:15:28.420 --> 00:15:31.012 Now, this is a particularly important general results, which 00:15:31.012 --> 00:15:34.468 I'd like to get very familiar with because it's going to crop 00:15:34.468 --> 00:15:38.500 up over and over again, and you want this sort of thing at your 00:15:38.500 --> 00:15:41.956 fingertips. For example, if I ask you to integrate one over X 00:15:41.956 --> 00:15:45.700 plus one with respect to XI, want you to be able to almost 00:15:45.700 --> 00:15:49.444 just write these things down if we identify the one over X plus 00:15:49.444 --> 00:15:53.764 one with the one over X Plus B, then we see that a is one. 00:15:54.570 --> 00:15:55.570 And B is one. 00:15:56.860 --> 00:16:02.185 Got a is one and be as one. This general result will give us one 00:16:02.185 --> 00:16:06.090 over one which is just one natural logarithm of the modulus 00:16:06.090 --> 00:16:10.350 of X Plus B, which in this case is X plus one. 00:16:14.280 --> 00:16:18.063 So our integral of one over X Plus One is just the logarithm 00:16:18.063 --> 00:16:22.672 of the denominator. Another example, the integral of one 00:16:22.672 --> 00:16:26.368 over 3X minus two with respect to X. 00:16:27.850 --> 00:16:31.282 Well, using our general result we see that a is 3. 00:16:33.720 --> 00:16:38.140 Be is minus two. Put all these into the formula and we'll get 00:16:38.140 --> 00:16:39.160 one over A. 00:16:39.800 --> 00:16:42.212 Play being 3 means we get one over 3. 00:16:43.740 --> 00:16:48.052 Natural logarithm. Of the modulus of the quantity we 00:16:48.052 --> 00:16:50.124 started with, which was 3X minus 2. 00:16:52.830 --> 00:16:55.910 Let's see, as I say, this is a particularly important 00:16:55.910 --> 00:16:58.990 result and you should have it at your fingertips. You'll 00:16:58.990 --> 00:17:02.686 need it over and over again in lots of situations as you 00:17:02.686 --> 00:17:03.302 perform integration. 00:17:05.180 --> 00:17:09.151 All the examples that we've looked at so far have been 00:17:09.151 --> 00:17:12.039 examples of indefinite integrals. They've not been any 00:17:12.039 --> 00:17:15.649 limits on the integral whatsoever, so let's now have a 00:17:15.649 --> 00:17:19.981 look at how we do an integration by substitution when there are 00:17:19.981 --> 00:17:24.674 limits on the integral as well, and the example that I want to 00:17:24.674 --> 00:17:29.367 look at is this one. Suppose we want to integrate from X equals 00:17:29.367 --> 00:17:32.255 1 to X equals 3, the function 9 00:17:32.255 --> 00:17:37.228 plus X. All raised to the power two and we want to integrate 00:17:37.228 --> 00:17:38.768 that with respect to X. 00:17:41.650 --> 00:17:46.660 Now have this just being an X to the power two. There would be no 00:17:46.660 --> 00:17:50.668 problem. Increase the power by one divided by the new power and 00:17:50.668 --> 00:17:55.344 you be finished. The problem is the 9 plus X. So as before we 00:17:55.344 --> 00:17:59.018 make a substitution, we make a substitution to try to simplify 00:17:59.018 --> 00:18:02.692 what we're doing and the substitution we make is U is 00:18:02.692 --> 00:18:04.362 equal to 9 plus X. 00:18:07.580 --> 00:18:09.708 Let's see what that does to the integral. 00:18:13.030 --> 00:18:17.450 Our integral will become, well, the 9 plus X becomes U so will 00:18:17.450 --> 00:18:18.470 have you squared. 00:18:21.490 --> 00:18:27.626 Let's deal next with the DX, as we've seen already, do you is 00:18:27.626 --> 00:18:29.514 equal to du DX? 00:18:31.670 --> 00:18:35.311 Multiplied by DX and in this particular case, this is a 00:18:35.311 --> 00:18:38.290 nice simple case. Du DX will be just one. 00:18:41.860 --> 00:18:45.765 So whenever we see a DX, we can replace it immediately 00:18:45.765 --> 00:18:49.670 by du to USD X, so this DX becomes a deal. 00:18:51.790 --> 00:18:55.417 So far it's the same as before, but now we've got to deal 00:18:55.417 --> 00:18:57.649 specifically with these limits. We have these limits. 00:18:59.260 --> 00:19:02.874 A lower limit of one, an upper limit of three, and these are 00:19:02.874 --> 00:19:04.264 limits on the variable X. 00:19:05.170 --> 00:19:10.149 We're integrating from X equals 1 to X equals 3. When we change 00:19:10.149 --> 00:19:13.885 the variable. It's very important that we changed the 00:19:13.885 --> 00:19:18.575 limits as well, and the way we do that is we use the given 00:19:18.575 --> 00:19:22.260 substitution. We use this to get new limits limits on you. 00:19:23.690 --> 00:19:26.080 Now when X is one. 00:19:27.010 --> 00:19:30.420 The lower limit you is going to be 9 + 1. 00:19:31.540 --> 00:19:34.900 10 so when X is one US 10. 00:19:37.560 --> 00:19:39.808 At the upper limit, when X is 3. 00:19:41.830 --> 00:19:45.016 Our substitution tells us that you is going to 00:19:45.016 --> 00:19:47.494 be 9 + 3, which is 12. 00:19:49.280 --> 00:19:52.439 We're going to use the substitution to change from 00:19:52.439 --> 00:19:54.896 limits on X to limits on you. 00:19:56.640 --> 00:19:59.335 So these limits on you are now 00:19:59.335 --> 00:20:02.922 you Ekwe. Tools 10 to you 00:20:02.922 --> 00:20:06.500 equals 12. And I've explicitly written the 00:20:06.500 --> 00:20:10.340 variable in here so that we know we've changed from X to 00:20:10.340 --> 00:20:10.660 you. 00:20:12.410 --> 00:20:15.632 This is straightforward to finish off because the integral 00:20:15.632 --> 00:20:20.286 of you squared is you cubed over 3 plus, plus no constant of 00:20:20.286 --> 00:20:23.508 integration because it's a definite integral and we write 00:20:23.508 --> 00:20:24.940 square brackets around here. 00:20:26.180 --> 00:20:28.980 And we write the limits on the right hand side. 00:20:31.770 --> 00:20:36.658 We finish this off by putting the upper limit in first, so you 00:20:36.658 --> 00:20:38.914 being 12 we want 12 cubed. 00:20:39.740 --> 00:20:40.928 Divided by three. 00:20:41.990 --> 00:20:45.182 That's the upper limit gone in. We want to put the lower 00:20:45.182 --> 00:20:47.842 limit in you being 10, so we want 10 cubed. 00:20:49.020 --> 00:20:52.850 Divided by three and as usual, we find the difference 00:20:52.850 --> 00:20:54.382 of these two quantities. 00:20:56.630 --> 00:20:58.058 Now 12 cubed. 00:20:58.780 --> 00:21:00.988 Get your Calculator out 12 cubed. 00:21:03.800 --> 00:21:11.504 Is 1728 so we have 1728 / 3 subtract 10 cubed which 00:21:11.504 --> 00:21:17.464 is 1000. Divided by three, so our final answer is 00:21:17.464 --> 00:21:19.356 going to be 728. 00:21:21.440 --> 00:21:22.520 Divided by three. 00:21:24.390 --> 00:21:28.090 And that's our first example of a definite integral using 00:21:28.090 --> 00:21:31.420 integration by substitution. Always remember that you use the 00:21:31.420 --> 00:21:35.490 given substitution to change the limits on the integral as well. 00:21:35.490 --> 00:21:37.340 Don't forget to do that. 00:21:42.640 --> 00:21:46.312 OK, all the examples that we've looked at so far have been 00:21:46.312 --> 00:21:48.454 fairly straightforward, and they've required a substitution 00:21:48.454 --> 00:21:52.126 of the form you equals X plus be a linear substitution. We're 00:21:52.126 --> 00:21:54.574 going to now have a look at some 00:21:54.574 --> 00:21:57.902 more complicated ones. All the examples I'm going to 00:21:57.902 --> 00:21:59.444 look at or of this form. 00:22:00.830 --> 00:22:06.526 So all going to take the form of the integral of F of G of X. 00:22:08.130 --> 00:22:09.558 G dash devex. 00:22:11.370 --> 00:22:14.672 DX now this looks horrendous. So what will do is will try and 00:22:14.672 --> 00:22:17.720 take it apart a little bit to see what's going on here. 00:22:19.270 --> 00:22:24.158 Suppose we have a function G of X which is equal to 1 00:22:24.158 --> 00:22:25.286 plus X squared. 00:22:28.780 --> 00:22:31.730 That's going to be this function in here, and you'll 00:22:31.730 --> 00:22:35.270 notice from this expression that the G of X is used as 00:22:35.270 --> 00:22:36.745 input to another function F. 00:22:37.840 --> 00:22:39.628 Suppose the function F is the 00:22:39.628 --> 00:22:42.605 square root function. The function which takes the 00:22:42.605 --> 00:22:45.938 square root of the input. So I'm going to write that 00:22:45.938 --> 00:22:49.574 like this. Suppose F of you is the square root of you. 00:22:51.430 --> 00:22:56.170 Then what does this mean? F of G of X? Well, this is called a 00:22:56.170 --> 00:22:59.014 composite function or the composition of the functions F&G 00:22:59.014 --> 00:23:04.070 the Function G is used as the input to F, so F of G of X. 00:23:05.710 --> 00:23:08.834 Well, the F function square roots the input, so we want 00:23:08.834 --> 00:23:12.526 to square root the input which is G of X, which is one 00:23:12.526 --> 00:23:13.378 plus X squared. 00:23:15.950 --> 00:23:18.560 So in this particular case, we're looking at an example 00:23:18.560 --> 00:23:21.692 that's going to be something like this, but the integral of F 00:23:21.692 --> 00:23:22.736 of G of X. 00:23:23.940 --> 00:23:27.765 Is the square root of 1 plus X squared. 00:23:29.520 --> 00:23:34.070 What about the G dash of XG dash trivex is the derivative of G 00:23:34.070 --> 00:23:35.370 with respect to X. 00:23:36.230 --> 00:23:40.689 Well, his GG of X is one plus X squared. What's its derivative? 00:23:41.820 --> 00:23:43.750 The derivative G Dash Devex. 00:23:46.230 --> 00:23:47.898 Is just 2X. 00:23:49.680 --> 00:23:53.554 So if I substitute for G, Dash, Devex as 2X, I'll be dealing 00:23:53.554 --> 00:23:55.044 with an integral like that. 00:23:56.920 --> 00:24:01.041 So as a first example, we're going to look at an integral of 00:24:01.041 --> 00:24:05.162 this particular form, and I hope that you can now see it's of 00:24:05.162 --> 00:24:09.283 this family of integrals. We've got a function G of X in here, 00:24:09.283 --> 00:24:12.770 one plus X squared, that's that, bit its derivative G dash, 00:24:12.770 --> 00:24:16.891 devex. The two X appears out here and the G itself is input 00:24:16.891 --> 00:24:20.061 to another function, which is the square routing function F. 00:24:20.910 --> 00:24:23.560 So it looks a bit complicated, but I hope we 00:24:23.560 --> 00:24:26.210 can see what all the all the different ingredients are. 00:24:27.220 --> 00:24:32.722 Now, the way we tackle a problem like this is to always make the 00:24:32.722 --> 00:24:37.438 substitution you equals G of X whatever the G of X was. 00:24:38.440 --> 00:24:42.710 Well, in this particular case, G of X was one plus X squared, so 00:24:42.710 --> 00:24:45.760 I'm going to make this substitution. You equals 1 plus 00:24:45.760 --> 00:24:49.396 X squared. Let's see what that will do to this integral. 00:24:52.940 --> 00:24:55.822 Letting you be one plus X squared will just have the 00:24:55.822 --> 00:24:57.394 square root of you in here. 00:25:01.710 --> 00:25:06.416 Now we've got it handled all this. The two X DX. Well in 00:25:06.416 --> 00:25:10.760 terms of Differentials, we know already that du is du DX DX. 00:25:10.760 --> 00:25:13.656 Well do you DX in this case is 00:25:13.656 --> 00:25:17.870 just 2X. So do you. DX DX is 2X DX. 00:25:19.230 --> 00:25:22.180 Now this is very fortunate because we see that the 00:25:22.180 --> 00:25:23.655 whole of two X DX. 00:25:24.900 --> 00:25:26.208 Can be replaced by. 00:25:27.380 --> 00:25:31.488 Do you so the whole of two X DX becomes the du and 00:25:31.488 --> 00:25:34.648 integrals of this form? This will always happen when we 00:25:34.648 --> 00:25:37.492 make a substitution like this. Now this is very 00:25:37.492 --> 00:25:40.020 straightforward to finish, provided you know a little 00:25:40.020 --> 00:25:43.812 bit of algebra. The square root of U is due to the 00:25:43.812 --> 00:25:46.656 power half, so we're integrating you to the half 00:25:46.656 --> 00:25:47.920 with respect to you. 00:25:49.190 --> 00:25:50.320 How do we do this? 00:25:51.660 --> 00:25:53.400 We increase the power by one. 00:25:54.660 --> 00:25:57.740 That will give us you to the half plus one that's 00:25:57.740 --> 00:25:59.420 1 1/2 or three over 2. 00:26:01.270 --> 00:26:05.288 And we divide by the gnu power and the new power. In this case 00:26:05.288 --> 00:26:09.593 is 3 over 2, so we're dividing by three over 2, and we add a 00:26:09.593 --> 00:26:10.454 constant of integration. 00:26:12.310 --> 00:26:16.434 Nearly finished. Dividing by three over 2 is like multiplying 00:26:16.434 --> 00:26:18.625 by 2/3, so I'll write that as 00:26:18.625 --> 00:26:21.485 2/3. And in terms of our 00:26:21.485 --> 00:26:25.220 original variable. You was one plus X squared. 00:26:28.180 --> 00:26:30.988 And all that needs to be raised to the power three over 2. 00:26:32.730 --> 00:26:34.319 And we need a constant of integration. 00:26:37.340 --> 00:26:40.370 So that's the first example of a much more complicated 00:26:40.370 --> 00:26:44.006 integral, and I want to go back and just point a few 00:26:44.006 --> 00:26:46.733 things out. Remember, a crucial step was to recognize 00:26:46.733 --> 00:26:50.066 that the 2X in here was the derivative of this quantity, 00:26:50.066 --> 00:26:51.884 one plus X squared in there. 00:26:54.019 --> 00:26:58.051 The G dash of X in the general case is the derivative of the 00:26:58.051 --> 00:26:59.779 input to the F function here. 00:27:01.009 --> 00:27:03.409 What will do is will have a look at another example and 00:27:03.409 --> 00:27:05.809 then we'll try and start to do some of this purely by 00:27:05.809 --> 00:27:08.409 inspection because at the end of the day we want you to be 00:27:08.409 --> 00:27:10.609 able to get to the stage where you're happy, almost just 00:27:10.609 --> 00:27:11.609 recognizing some of these integrals. 00:27:17.789 --> 00:27:20.469 OK, let's have another look at another example 00:27:20.469 --> 00:27:23.149 of one of these complicated things. F. Of 00:27:23.149 --> 00:27:24.154 G. Of X. 00:27:26.099 --> 00:27:27.407 G dash devex. 00:27:30.919 --> 00:27:36.679 And in this case I'm going to take as my G of X function. The 00:27:36.679 --> 00:27:41.671 following G of X is going to be 2 X squared plus one. 00:27:44.939 --> 00:27:48.929 This is going to be used as input to the F function, and in 00:27:48.929 --> 00:27:52.919 this case I'm going to choose F as this function F of you is 00:27:52.919 --> 00:27:56.339 going to be the function which takes an input, finds it square 00:27:56.339 --> 00:27:59.759 roots, and then finds its reciprocal. So F of you is one 00:27:59.759 --> 00:28:01.469 over the square root of you. 00:28:03.859 --> 00:28:07.079 What would that look like in terms of this integral? 00:28:07.079 --> 00:28:10.299 Well, this integral will be F of G of X. 00:28:11.899 --> 00:28:16.043 All this means that the G of X function is used as input to 00:28:16.043 --> 00:28:20.187 F. So if we use two X squared as one as input to this 00:28:20.187 --> 00:28:24.035 function, F will get one over the square root of 2 X squared 00:28:24.035 --> 00:28:26.403 plus one. So I'll be integrating something like 00:28:26.403 --> 00:28:26.699 this. 00:28:34.449 --> 00:28:36.689 We also need to look at the G Dash Devex. 00:28:38.189 --> 00:28:42.869 Well, because G of X is 2 X squared plus one, if 00:28:42.869 --> 00:28:45.989 we differentiate GG dash, devex will be just 00:28:45.989 --> 00:28:47.939 simply to choose of 4X. 00:28:49.929 --> 00:28:53.369 So for G Dash Devex, I'm going to write 4X. 00:28:54.519 --> 00:28:55.487 And with the DX. 00:28:57.049 --> 00:29:00.052 Look in this example. It's an integral like this now this 00:29:00.052 --> 00:29:03.055 looks very complicated. It might have actually been given to you 00:29:03.055 --> 00:29:06.331 in a form that looked more like this because we might have 00:29:06.331 --> 00:29:10.153 written the 1 * 4 X all as the single term. We might have 00:29:10.153 --> 00:29:13.702 actually written it down like 4X divided by the square root of 2 00:29:13.702 --> 00:29:16.432 X squared at one, all integrated with respect to X. 00:29:18.809 --> 00:29:21.955 Just look a bit complicated, but we'll see by making a 00:29:21.955 --> 00:29:24.529 substitution that we can make some progress and the 00:29:24.529 --> 00:29:27.961 substitution that will make as before is we let you be the 00:29:27.961 --> 00:29:29.105 function G of X. 00:29:30.219 --> 00:29:32.307 We let you be this function in here. 00:29:33.499 --> 00:29:34.549 So if you. 00:29:35.979 --> 00:29:39.464 Is G of X or G of X? In this case was two X squared plus one. 00:29:42.199 --> 00:29:43.427 What's going to happen? 00:29:44.169 --> 00:29:46.297 Well, let's put all this into this integral. 00:29:47.559 --> 00:29:51.387 Don't worry about the 4X for a minute, but the two X 00:29:51.387 --> 00:29:54.896 squared plus one here in the denominator becomes a U, so 00:29:54.896 --> 00:29:58.086 we'll have a square root of you in the denominator. 00:29:59.529 --> 00:30:04.482 We've got to. Look after the four X DX as well, but in terms 00:30:04.482 --> 00:30:09.071 of differentials do you remember is du DX, which is 4X times DX 00:30:09.071 --> 00:30:13.307 and you'll see because of the nature of this sort of problem 00:30:13.307 --> 00:30:17.896 that the four X DX quantity is exactly what we've got over here 00:30:17.896 --> 00:30:23.544 for X DX. So the whole of four X DX becomes du the whole of that. 00:30:25.029 --> 00:30:25.977 Becomes a du. 00:30:27.549 --> 00:30:29.988 That's the nice thing about problems like this. That 00:30:29.988 --> 00:30:32.698 substitution will always make them drop out in this way. 00:30:34.539 --> 00:30:38.007 Again, provided that you know a bit of algebra, you can finish 00:30:38.007 --> 00:30:42.053 this off. Let's just do that. The square root of U is due to 00:30:42.053 --> 00:30:46.619 the power half. One over the square root will give us you to 00:30:46.619 --> 00:30:49.864 the minus 1/2. So the problem that we're integrating here is 00:30:49.864 --> 00:30:51.339 you to the minus 1/2. 00:30:52.579 --> 00:30:53.699 With respect to you. 00:30:56.409 --> 00:31:00.069 You to the minus 1/2 integrated we increase the power by one. 00:31:00.779 --> 00:31:05.015 So we increase minus 1/2 by one will give us plus 1/2. 00:31:07.229 --> 00:31:09.287 And we divide by the new power. 00:31:10.979 --> 00:31:14.367 And we need a constant of integration and for all intents 00:31:14.367 --> 00:31:17.447 and purposes that's finished, but we can revert to our 00:31:17.447 --> 00:31:20.527 original variable X through the substitution. What did we have? 00:31:21.419 --> 00:31:22.448 We had you. 00:31:23.669 --> 00:31:29.073 What's 2 X squared plus one? So this is going to become two X 00:31:29.073 --> 00:31:33.705 squared plus one all raised to the power half plus C and 00:31:33.705 --> 00:31:37.951 division by 1/2 is like multiplication by two, so I have 00:31:37.951 --> 00:31:42.583 a two there at the beginning and that's the solution of this 00:31:42.583 --> 00:31:45.285 fairly complicated integral done through a substitution. 00:31:52.749 --> 00:31:56.397 So let me try and extract from all that are fairly general 00:31:56.397 --> 00:32:00.045 results and the general result is this that if we have an 00:32:00.045 --> 00:32:02.173 integral of F of G of X. 00:32:03.319 --> 00:32:06.089 He dashed of X DX. 00:32:08.379 --> 00:32:10.147 Then the substitution U 00:32:10.147 --> 00:32:11.915 equals G of X. 00:32:13.419 --> 00:32:18.326 Do you is G dashed of XDX? 00:32:19.349 --> 00:32:21.665 That substitution will always transform this 00:32:21.665 --> 00:32:26.683 integral into simply F of G of X, which is F of U. 00:32:29.059 --> 00:32:31.299 And the whole of G dash DX. 00:32:32.379 --> 00:32:33.969 Will become simply do youth. 00:32:36.049 --> 00:32:39.352 And this hopefully will be an integral which is 00:32:39.352 --> 00:32:40.820 much simpler to evaluate. 00:32:42.109 --> 00:32:45.124 Now really what we want to be able to do is get you into the 00:32:45.124 --> 00:32:47.737 habit of spotting some of these and being able to write down the 00:32:47.737 --> 00:32:50.350 answer straight away. So let me just give you one or two more 00:32:50.350 --> 00:32:52.360 examples where hopefully we can actually spot what's going on. 00:32:52.929 --> 00:32:56.878 Supposing we looking at the integral of E to the X 00:32:56.878 --> 00:33:00.468 squared multiplied by two X and we want to integrate 00:33:00.468 --> 00:33:02.263 that with respect to X. 00:33:03.639 --> 00:33:07.359 Let's try and compare what we've got here with what's up here. 00:33:09.259 --> 00:33:13.666 If you think of the X squared as being the G of X. 00:33:15.019 --> 00:33:17.819 So this bit is the G of X. 00:33:21.089 --> 00:33:24.953 If we differentiate, X squared would get 2X and you see that 00:33:24.953 --> 00:33:29.139 appears out here, so this, but in here is the G Dash, Devex. 00:33:31.109 --> 00:33:33.908 And there's another function involved in here as well. 00:33:33.908 --> 00:33:37.018 It's the F function, and in this particular example, the 00:33:37.018 --> 00:33:40.439 F function is the exponential function, and we see that G 00:33:40.439 --> 00:33:43.860 is input to F because X squared is input to the 00:33:43.860 --> 00:33:44.482 exponential function. 00:33:45.759 --> 00:33:49.269 So if we want to tackle this particular integral, the 00:33:49.269 --> 00:33:53.481 substitution U equals whatever the G of X was, which in this 00:33:53.481 --> 00:33:55.587 case is X squared will simplify 00:33:55.587 --> 00:34:01.519 this integral. You being X squared do you will be 2X DX. 00:34:02.289 --> 00:34:05.835 And immediately this integral will become the integral of. 00:34:07.329 --> 00:34:11.164 E to the power X squared, which was E to the power you. 00:34:12.479 --> 00:34:18.108 And automatically the two X DX is taken care of in the du. 00:34:20.809 --> 00:34:24.150 Now this is just the integral of the exponential function E to EU 00:34:24.150 --> 00:34:27.234 with respect to you and we can write the answer straight down 00:34:27.234 --> 00:34:30.318 as the same thing eater, the you and a constant of integration. 00:34:31.639 --> 00:34:35.755 If we return to our original variables, you was X squared, so 00:34:35.755 --> 00:34:40.557 will hav E to the X squared plus a constant. So this quantity E 00:34:40.557 --> 00:34:45.702 to the X squared plus a constant is the integral of two XE to the 00:34:45.702 --> 00:34:46.731 X squared DX. 00:34:47.789 --> 00:34:53.081 And as I say, we want to get into the habit of being able 00:34:53.081 --> 00:34:57.239 to almost spotless, and you should get into the habit of 00:34:57.239 --> 00:35:01.019 spotting that the quantity here the two X is the 00:35:01.019 --> 00:35:04.043 derivative of this function in this other composite 00:35:04.043 --> 00:35:07.823 function. Here, let's see if we can do one straightaway. 00:35:07.823 --> 00:35:11.225 Supposing, supposing that I ask you to integrate the 00:35:11.225 --> 00:35:16.139 cosine of three X to the power 4 * 12 X cubed. Suppose 00:35:16.139 --> 00:35:19.163 we want to integrate this horrible looking thing. 00:35:20.229 --> 00:35:24.103 Now the thing I want you to spot is that if you differentiate 00:35:24.103 --> 00:35:27.977 this function in here 3X to the power four, you'd actually get 4 00:35:27.977 --> 00:35:31.553 threes or 12X cubed. You get the concert if it's out here. 00:35:32.229 --> 00:35:34.357 So this is like the G of X. 00:35:35.779 --> 00:35:39.310 And this is like the G Dash, Devex, and making the 00:35:39.310 --> 00:35:42.199 substitution you equals 3X plus four would immediately reduce 00:35:42.199 --> 00:35:46.372 you to a much simpler integral, just in terms of you. I'm hoping 00:35:46.372 --> 00:35:50.545 that you by now you can spot that if we integrate, this will 00:35:50.545 --> 00:35:52.150 actually just get the sign. 00:35:53.859 --> 00:35:57.579 Of three X to the four plus a constant of integration. Suggest 00:35:57.579 --> 00:36:01.919 you go back and look at that again if you not too happy about 00:36:01.919 --> 00:36:05.029 it. And that's integration by substitution.