WEBVTT 00:00:01.510 --> 00:00:02.800 We already know. 00:00:03.960 --> 00:00:08.420 When we differentiate log X. 00:00:09.430 --> 00:00:10.770 We end up with. 00:00:13.870 --> 00:00:15.938 Is one over X? 00:00:17.230 --> 00:00:24.742 We also know that if we've got Y equals log of a 00:00:24.742 --> 00:00:26.620 function of X. 00:00:27.420 --> 00:00:29.060 And we differentiate it. 00:00:32.170 --> 00:00:37.656 Then what we end up with is the derivative of that function over 00:00:37.656 --> 00:00:39.344 the function of X. 00:00:40.520 --> 00:00:44.557 Now the point about integrating is if we can recognize something 00:00:44.557 --> 00:00:48.227 that's a differential, then we can simply reverse the process. 00:00:48.227 --> 00:00:52.631 So what we're going to be looking for or looking at in 00:00:52.631 --> 00:00:56.301 this case, is functions that look like this that require 00:00:56.301 --> 00:00:59.971 integration, so we can go back from there to there. 00:01:01.910 --> 00:01:05.582 So let's see if we can just write that little bit down 00:01:05.582 --> 00:01:08.336 again and then have a look at some examples. 00:01:09.900 --> 00:01:16.860 So no, that is why is the log of a function of 00:01:16.860 --> 00:01:23.240 X then divide by The X is the derivative of that 00:01:23.240 --> 00:01:26.140 function divided by the function. 00:01:27.870 --> 00:01:31.398 So therefore, if we can recognize. 00:01:33.920 --> 00:01:35.250 That form. 00:01:38.110 --> 00:01:43.030 And we want to integrate it. Then we can claim straight away 00:01:43.030 --> 00:01:48.770 that this is the log of the function of X plus. Of course a 00:01:48.770 --> 00:01:52.460 constant of integration because there are no limits here. 00:01:53.420 --> 00:01:57.190 So we're going to be looking for this. We're going to be looking 00:01:57.190 --> 00:02:00.380 at what we've been given to integrate and can we spot? 00:02:01.460 --> 00:02:05.560 A derivative. Over the function, or something 00:02:05.560 --> 00:02:06.970 approaching a derivative. 00:02:08.050 --> 00:02:11.625 So now we've got the result. Let's look at some examples. 00:02:12.980 --> 00:02:19.124 So we take the integral of tan XDX. 00:02:19.980 --> 00:02:24.900 Now it doesn't look much like one of the examples. We've just 00:02:24.900 --> 00:02:30.230 been talking about, but we know that we can redefine Tan X sign 00:02:30.230 --> 00:02:31.870 X over cause X. 00:02:33.220 --> 00:02:38.186 And now when we look at the derivative of cars is minus sign 00:02:38.186 --> 00:02:43.014 so. The numerator is very nearly the derivative of the 00:02:43.014 --> 00:02:48.126 denominator, so let's make it so. Let's put in minus sign X. 00:02:51.720 --> 00:02:54.970 Now having putting the minus sign, we've achieved what we 00:02:54.970 --> 00:02:57.895 want. The numerator is the derivative of the denominator, 00:02:57.895 --> 00:03:00.170 the top is the derivative of the 00:03:00.170 --> 00:03:05.302 bottom. But we need to put in that balancing minus sign so 00:03:05.302 --> 00:03:08.740 that we can retain the equality of these two 00:03:08.740 --> 00:03:12.178 expressions. Having done that, we can now write this 00:03:12.178 --> 00:03:12.560 down. 00:03:13.860 --> 00:03:20.685 Minus and it's that minus sign. The log of caused X plus a 00:03:20.685 --> 00:03:22.785 constant of integration, see. 00:03:24.510 --> 00:03:29.200 We're subtracting a log, which means we're dividing by what's 00:03:29.200 --> 00:03:30.607 within the log. 00:03:31.750 --> 00:03:36.106 Function, so we're dividing by cause what we do know is 00:03:36.106 --> 00:03:39.670 with. With dividing by cars, then that's one over 00:03:39.670 --> 00:03:44.818 cars and that sank. So this is log of sex X Plus C. 00:03:50.520 --> 00:03:54.540 Now let's go on again and have a look at another example. 00:03:56.460 --> 00:04:01.590 Integral of X over one plus 00:04:01.590 --> 00:04:04.155 X squared DX. 00:04:06.250 --> 00:04:10.921 Look at the bottom and differentiate it. Its derivative 00:04:10.921 --> 00:04:17.668 is 2X only got X on top, that's no problem. Let's make it 00:04:17.668 --> 00:04:21.301 2X on top by multiplying by two. 00:04:22.820 --> 00:04:25.977 If we've multiplied by two, we've got to divide by two, 00:04:25.977 --> 00:04:29.134 and that means we want a half of that result there. 00:04:29.134 --> 00:04:32.578 So now this is balanced out and it's the same as that. 00:04:34.150 --> 00:04:38.451 What we've got on the top now is very definitely the 00:04:38.451 --> 00:04:41.970 derivative of what's on the bottom, so again, we 00:04:41.970 --> 00:04:47.053 can have a half the log of one plus X squared plus C. 00:04:48.360 --> 00:04:53.810 We can even have this with look like very complicated 00:04:53.810 --> 00:04:59.805 functions, so let's have a look at one over X Times 00:04:59.805 --> 00:05:02.530 the natural log of X. 00:05:07.780 --> 00:05:12.916 Doesn't look like what we've got does it? But let's remember that 00:05:12.916 --> 00:05:19.336 the derivative of log X is one over X. So if I write this a 00:05:19.336 --> 00:05:24.044 little bit differently, one over X divided by log X DX. 00:05:25.590 --> 00:05:30.738 Then we can see that what's on top is indeed the derivative of 00:05:30.738 --> 00:05:35.490 what's on the bottom, and so, again, this is the log of. 00:05:37.000 --> 00:05:43.000 Log of X plus a constant of integration. See, so even in 00:05:43.000 --> 00:05:48.500 something like that we can find what it is we're actually 00:05:48.500 --> 00:05:51.500 looking for. Let's take one more 00:05:51.500 --> 00:05:57.316 example. A little bit contrived, but it does show you how you 00:05:57.316 --> 00:06:00.158 need to work and look to see. 00:06:01.720 --> 00:06:03.508 If what you've got on the. 00:06:04.220 --> 00:06:08.684 Numerator is in fact the derivative of the denominator. 00:06:14.270 --> 00:06:15.929 So let's have a look at this. 00:06:18.340 --> 00:06:23.160 Looks quite fearsome as it's written, but let's just think 00:06:23.160 --> 00:06:29.426 about what we would get if we differentiate it X Sign X. I'll 00:06:29.426 --> 00:06:36.174 just do that over here. Let's say Y equals X sign X. Now this 00:06:36.174 --> 00:06:43.886 is a product, it's a U times by AV, so we know that if Y equals 00:06:43.886 --> 00:06:46.296 UV when we do the 00:06:46.296 --> 00:06:53.590 differentiation. Why by DX is UDV by the X 00:06:53.590 --> 00:06:56.790 plus VDU by DX? 00:06:57.720 --> 00:07:04.908 So in this case you is X&V is synex, so that's X 00:07:04.908 --> 00:07:09.101 cause X Plus V which is synex. 00:07:11.350 --> 00:07:16.756 Times by du by DX, but you was X or do you buy X is just one? 00:07:17.370 --> 00:07:20.202 So if we look what we can see 00:07:20.202 --> 00:07:23.220 here. Is that the numerator? 00:07:23.910 --> 00:07:29.562 Is X cause X sign X, which is the derivative of the 00:07:29.562 --> 00:07:34.272 denominator X sign X, and so again complicated though it 00:07:34.272 --> 00:07:39.453 looks we've been able to spot that the numerator is again 00:07:39.453 --> 00:07:44.163 the derivative of the denominator, and so we can say 00:07:44.163 --> 00:07:49.815 straight away that the result of this integral is the log of 00:07:49.815 --> 00:07:52.170 the denominator X Sign X. 00:07:54.380 --> 00:07:58.219 Sometimes you have to look very closely and let's just remember 00:07:58.219 --> 00:08:02.407 if we just look back at this one. But sometimes you might 00:08:02.407 --> 00:08:06.944 have to balance the function in order to be able to make it 00:08:06.944 --> 00:08:11.132 look like you want it to look. But quite often it's fairly 00:08:11.132 --> 00:08:15.669 clear that that's what you have to do. So do remember, this is 00:08:15.669 --> 00:08:19.159 a very typical standard form of integration of very important 00:08:19.159 --> 00:08:22.998 one, and one that occurs a great deal when looking at 00:08:22.998 --> 00:08:23.696 differential equations.