• 0:02 - 0:03
• 0:04 - 0:08
When we differentiate
log X.
• 0:09 - 0:11
We end up with.
• 0:14 - 0:16
Is one over X?
• 0:17 - 0:25
We also know that if we've
got Y equals log of a
• 0:25 - 0:27
function of X.
• 0:27 - 0:29
And we differentiate it.
• 0:32 - 0:38
Then what we end up with is the
derivative of that function over
• 0:38 - 0:39
the function of X.
• 0:41 - 0:45
is if we can recognize something
• 0:45 - 0:48
that's a differential, then we
can simply reverse the process.
• 0:48 - 0:53
So what we're going to be
looking for or looking at in
• 0:53 - 0:56
this case, is functions that
look like this that require
• 0:56 - 1:00
integration, so we can go back
from there to there.
• 1:02 - 1:06
So let's see if we can just
write that little bit down
• 1:06 - 1:08
again and then have a look
at some examples.
• 1:10 - 1:17
So no, that is why is
the log of a function of
• 1:17 - 1:23
X then divide by The X
is the derivative of that
• 1:23 - 1:26
function divided by the
function.
• 1:28 - 1:31
So therefore, if
we can recognize.
• 1:34 - 1:35
That form.
• 1:38 - 1:43
And we want to integrate it.
Then we can claim straight away
• 1:43 - 1:49
that this is the log of the
function of X plus. Of course a
• 1:49 - 1:52
constant of integration because
there are no limits here.
• 1:53 - 1:57
So we're going to be looking for
this. We're going to be looking
• 1:57 - 2:00
at what we've been given to
integrate and can we spot?
• 2:01 - 2:06
A derivative. Over the
function, or something
• 2:06 - 2:07
approaching a derivative.
• 2:08 - 2:12
So now we've got the result.
Let's look at some examples.
• 2:13 - 2:19
So we take the integral
of tan XDX.
• 2:20 - 2:25
Now it doesn't look much like
one of the examples. We've just
• 2:25 - 2:30
been talking about, but we know
that we can redefine Tan X sign
• 2:30 - 2:32
X over cause X.
• 2:33 - 2:38
And now when we look at the
derivative of cars is minus sign
• 2:38 - 2:43
so. The numerator is very nearly
the derivative of the
• 2:43 - 2:48
denominator, so let's make it
so. Let's put in minus sign X.
• 2:52 - 2:55
Now having putting the minus
sign, we've achieved what we
• 2:55 - 2:58
want. The numerator is the
derivative of the denominator,
• 2:58 - 3:00
the top is the derivative of the
• 3:00 - 3:05
bottom. But we need to put in
that balancing minus sign so
• 3:05 - 3:09
that we can retain the
equality of these two
• 3:09 - 3:12
expressions. Having done
that, we can now write this
• 3:12 - 3:13
down.
• 3:14 - 3:21
Minus and it's that minus sign.
The log of caused X plus a
• 3:21 - 3:23
constant of integration, see.
• 3:25 - 3:29
We're subtracting a log, which
means we're dividing by what's
• 3:29 - 3:31
within the log.
• 3:32 - 3:36
Function, so we're dividing
by cause what we do know is
• 3:36 - 3:40
with. With dividing by
cars, then that's one over
• 3:40 - 3:45
cars and that sank. So this
is log of sex X Plus C.
• 3:51 - 3:55
Now let's go on again and have a
look at another example.
• 3:56 - 4:02
Integral of X
over one plus
• 4:02 - 4:04
X squared DX.
• 4:06 - 4:11
Look at the bottom and
differentiate it. Its derivative
• 4:11 - 4:18
is 2X only got X on top,
that's no problem. Let's make it
• 4:18 - 4:21
2X on top by multiplying by two.
• 4:23 - 4:26
If we've multiplied by two,
we've got to divide by two,
• 4:26 - 4:29
and that means we want a
half of that result there.
• 4:29 - 4:33
So now this is balanced out
and it's the same as that.
• 4:34 - 4:38
What we've got on the top
now is very definitely the
• 4:38 - 4:42
derivative of what's on
the bottom, so again, we
• 4:42 - 4:47
can have a half the log of
one plus X squared plus C.
• 4:48 - 4:54
We can even have this with
look like very complicated
• 4:54 - 5:00
functions, so let's have a
look at one over X Times
• 5:00 - 5:03
the natural log of X.
• 5:08 - 5:13
Doesn't look like what we've got
does it? But let's remember that
• 5:13 - 5:19
the derivative of log X is one
over X. So if I write this a
• 5:19 - 5:24
little bit differently, one over
X divided by log X DX.
• 5:26 - 5:31
Then we can see that what's on
top is indeed the derivative of
• 5:31 - 5:35
what's on the bottom, and so,
again, this is the log of.
• 5:37 - 5:43
Log of X plus a constant of
integration. See, so even in
• 5:43 - 5:48
something like that we can find
what it is we're actually
• 5:48 - 5:52
looking for. Let's take one more
• 5:52 - 5:57
example. A little bit contrived,
but it does show you how you
• 5:57 - 6:00
need to work and look to see.
• 6:02 - 6:04
If what you've got on the.
• 6:04 - 6:09
Numerator is in fact the
derivative of the denominator.
• 6:14 - 6:16
So let's have a look at this.
• 6:18 - 6:23
Looks quite fearsome as it's
written, but let's just think
• 6:23 - 6:29
about what we would get if we
differentiate it X Sign X. I'll
• 6:29 - 6:36
just do that over here. Let's
say Y equals X sign X. Now this
• 6:36 - 6:44
is a product, it's a U times by
AV, so we know that if Y equals
• 6:44 - 6:46
UV when we do the
• 6:46 - 6:54
differentiation. Why by DX
is UDV by the X
• 6:54 - 6:57
plus VDU by DX?
• 6:58 - 7:05
So in this case you is
X&V is synex, so that's X
• 7:05 - 7:09
cause X Plus V which is
synex.
• 7:11 - 7:17
Times by du by DX, but you was X
or do you buy X is just one?
• 7:17 - 7:20
So if we look what we can see
• 7:20 - 7:23
here. Is that the numerator?
• 7:24 - 7:30
Is X cause X sign X, which is
the derivative of the
• 7:30 - 7:34
denominator X sign X, and so
again complicated though it
• 7:34 - 7:39
looks we've been able to spot
that the numerator is again
• 7:39 - 7:44
the derivative of the
denominator, and so we can say
• 7:44 - 7:50
straight away that the result
of this integral is the log of
• 7:50 - 7:52
the denominator X Sign X.
• 7:54 - 7:58
Sometimes you have to look very
closely and let's just remember
• 7:58 - 8:02
if we just look back at this
one. But sometimes you might
• 8:02 - 8:07
have to balance the function in
order to be able to make it
• 8:07 - 8:11
look like you want it to look.
But quite often it's fairly
• 8:11 - 8:16
clear that that's what you have
to do. So do remember, this is
• 8:16 - 8:19
a very typical standard form of
integration of very important
• 8:19 - 8:23
one, and one that occurs a
great deal when looking at
• 8:23 - 8:24
differential equations.
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