0:00:01.510,0:00:02.800 We already know. 0:00:03.960,0:00:08.420 When we differentiate[br]log X. 0:00:09.430,0:00:10.770 We end up with. 0:00:13.870,0:00:15.938 Is one over X? 0:00:17.230,0:00:24.742 We also know that if we've[br]got Y equals log of a 0:00:24.742,0:00:26.620 function of X. 0:00:27.420,0:00:29.060 And we differentiate it. 0:00:32.170,0:00:37.656 Then what we end up with is the[br]derivative of that function over 0:00:37.656,0:00:39.344 the function of X. 0:00:40.520,0:00:44.557 Now the point about integrating[br]is if we can recognize something 0:00:44.557,0:00:48.227 that's a differential, then we[br]can simply reverse the process. 0:00:48.227,0:00:52.631 So what we're going to be[br]looking for or looking at in 0:00:52.631,0:00:56.301 this case, is functions that[br]look like this that require 0:00:56.301,0:00:59.971 integration, so we can go back[br]from there to there. 0:01:01.910,0:01:05.582 So let's see if we can just[br]write that little bit down 0:01:05.582,0:01:08.336 again and then have a look[br]at some examples. 0:01:09.900,0:01:16.860 So no, that is why is[br]the log of a function of 0:01:16.860,0:01:23.240 X then divide by The X[br]is the derivative of that 0:01:23.240,0:01:26.140 function divided by the[br]function. 0:01:27.870,0:01:31.398 So therefore, if[br]we can recognize. 0:01:33.920,0:01:35.250 That form. 0:01:38.110,0:01:43.030 And we want to integrate it.[br]Then we can claim straight away 0:01:43.030,0:01:48.770 that this is the log of the[br]function of X plus. Of course a 0:01:48.770,0:01:52.460 constant of integration because[br]there are no limits here. 0:01:53.420,0:01:57.190 So we're going to be looking for[br]this. We're going to be looking 0:01:57.190,0:02:00.380 at what we've been given to[br]integrate and can we spot? 0:02:01.460,0:02:05.560 A derivative. Over the[br]function, or something 0:02:05.560,0:02:06.970 approaching a derivative. 0:02:08.050,0:02:11.625 So now we've got the result.[br]Let's look at some examples. 0:02:12.980,0:02:19.124 So we take the integral[br]of tan XDX. 0:02:19.980,0:02:24.900 Now it doesn't look much like[br]one of the examples. We've just 0:02:24.900,0:02:30.230 been talking about, but we know[br]that we can redefine Tan X sign 0:02:30.230,0:02:31.870 X over cause X. 0:02:33.220,0:02:38.186 And now when we look at the[br]derivative of cars is minus sign 0:02:38.186,0:02:43.014 so. The numerator is very nearly[br]the derivative of the 0:02:43.014,0:02:48.126 denominator, so let's make it[br]so. Let's put in minus sign X. 0:02:51.720,0:02:54.970 Now having putting the minus[br]sign, we've achieved what we 0:02:54.970,0:02:57.895 want. The numerator is the[br]derivative of the denominator, 0:02:57.895,0:03:00.170 the top is the derivative of the 0:03:00.170,0:03:05.302 bottom. But we need to put in[br]that balancing minus sign so 0:03:05.302,0:03:08.740 that we can retain the[br]equality of these two 0:03:08.740,0:03:12.178 expressions. Having done[br]that, we can now write this 0:03:12.178,0:03:12.560 down. 0:03:13.860,0:03:20.685 Minus and it's that minus sign.[br]The log of caused X plus a 0:03:20.685,0:03:22.785 constant of integration, see. 0:03:24.510,0:03:29.200 We're subtracting a log, which[br]means we're dividing by what's 0:03:29.200,0:03:30.607 within the log. 0:03:31.750,0:03:36.106 Function, so we're dividing[br]by cause what we do know is 0:03:36.106,0:03:39.670 with. With dividing by[br]cars, then that's one over 0:03:39.670,0:03:44.818 cars and that sank. So this[br]is log of sex X Plus C. 0:03:50.520,0:03:54.540 Now let's go on again and have a[br]look at another example. 0:03:56.460,0:04:01.590 Integral of X[br]over one plus 0:04:01.590,0:04:04.155 X squared DX. 0:04:06.250,0:04:10.921 Look at the bottom and[br]differentiate it. Its derivative 0:04:10.921,0:04:17.668 is 2X only got X on top,[br]that's no problem. Let's make it 0:04:17.668,0:04:21.301 2X on top by multiplying by two. 0:04:22.820,0:04:25.977 If we've multiplied by two,[br]we've got to divide by two, 0:04:25.977,0:04:29.134 and that means we want a[br]half of that result there. 0:04:29.134,0:04:32.578 So now this is balanced out[br]and it's the same as that. 0:04:34.150,0:04:38.451 What we've got on the top[br]now is very definitely the 0:04:38.451,0:04:41.970 derivative of what's on[br]the bottom, so again, we 0:04:41.970,0:04:47.053 can have a half the log of[br]one plus X squared plus C. 0:04:48.360,0:04:53.810 We can even have this with[br]look like very complicated 0:04:53.810,0:04:59.805 functions, so let's have a[br]look at one over X Times 0:04:59.805,0:05:02.530 the natural log of X. 0:05:07.780,0:05:12.916 Doesn't look like what we've got[br]does it? But let's remember that 0:05:12.916,0:05:19.336 the derivative of log X is one[br]over X. So if I write this a 0:05:19.336,0:05:24.044 little bit differently, one over[br]X divided by log X DX. 0:05:25.590,0:05:30.738 Then we can see that what's on[br]top is indeed the derivative of 0:05:30.738,0:05:35.490 what's on the bottom, and so,[br]again, this is the log of. 0:05:37.000,0:05:43.000 Log of X plus a constant of[br]integration. See, so even in 0:05:43.000,0:05:48.500 something like that we can find[br]what it is we're actually 0:05:48.500,0:05:51.500 looking for. Let's take one more 0:05:51.500,0:05:57.316 example. A little bit contrived,[br]but it does show you how you 0:05:57.316,0:06:00.158 need to work and look to see. 0:06:01.720,0:06:03.508 If what you've got on the. 0:06:04.220,0:06:08.684 Numerator is in fact the[br]derivative of the denominator. 0:06:14.270,0:06:15.929 So let's have a look at this. 0:06:18.340,0:06:23.160 Looks quite fearsome as it's[br]written, but let's just think 0:06:23.160,0:06:29.426 about what we would get if we[br]differentiate it X Sign X. I'll 0:06:29.426,0:06:36.174 just do that over here. Let's[br]say Y equals X sign X. Now this 0:06:36.174,0:06:43.886 is a product, it's a U times by[br]AV, so we know that if Y equals 0:06:43.886,0:06:46.296 UV when we do the 0:06:46.296,0:06:53.590 differentiation. Why by DX[br]is UDV by the X 0:06:53.590,0:06:56.790 plus VDU by DX? 0:06:57.720,0:07:04.908 So in this case you is[br]X&V is synex, so that's X 0:07:04.908,0:07:09.101 cause X Plus V which is[br]synex. 0:07:11.350,0:07:16.756 Times by du by DX, but you was X[br]or do you buy X is just one? 0:07:17.370,0:07:20.202 So if we look what we can see 0:07:20.202,0:07:23.220 here. Is that the numerator? 0:07:23.910,0:07:29.562 Is X cause X sign X, which is[br]the derivative of the 0:07:29.562,0:07:34.272 denominator X sign X, and so[br]again complicated though it 0:07:34.272,0:07:39.453 looks we've been able to spot[br]that the numerator is again 0:07:39.453,0:07:44.163 the derivative of the[br]denominator, and so we can say 0:07:44.163,0:07:49.815 straight away that the result[br]of this integral is the log of 0:07:49.815,0:07:52.170 the denominator X Sign X. 0:07:54.380,0:07:58.219 Sometimes you have to look very[br]closely and let's just remember 0:07:58.219,0:08:02.407 if we just look back at this[br]one. But sometimes you might 0:08:02.407,0:08:06.944 have to balance the function in[br]order to be able to make it 0:08:06.944,0:08:11.132 look like you want it to look.[br]But quite often it's fairly 0:08:11.132,0:08:15.669 clear that that's what you have[br]to do. So do remember, this is 0:08:15.669,0:08:19.159 a very typical standard form of[br]integration of very important 0:08:19.159,0:08:22.998 one, and one that occurs a[br]great deal when looking at 0:08:22.998,0:08:23.696 differential equations.