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.