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We already know.
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When we differentiate
log X.
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We end up with.
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Is one over X?
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We also know that if we've
got Y equals log of a
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function of X.
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And we differentiate it.
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Then what we end up with is the
derivative of that function over
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the function of X.
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Now the point about integrating
is if we can recognize something
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that's a differential, then we
can simply reverse the process.
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So what we're going to be
looking for or looking at in
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this case, is functions that
look like this that require
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integration, so we can go back
from there to there.
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So let's see if we can just
write that little bit down
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again and then have a look
at some examples.
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So no, that is why is
the log of a function of
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X then divide by The X
is the derivative of that
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function divided by the
function.
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So therefore, if
we can recognize.
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That form.
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And we want to integrate it.
Then we can claim straight away
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that this is the log of the
function of X plus. Of course a
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constant of integration because
there are no limits here.
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So we're going to be looking for
this. We're going to be looking
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at what we've been given to
integrate and can we spot?
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A derivative. Over the
function, or something
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approaching a derivative.
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So now we've got the result.
Let's look at some examples.
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So we take the integral
of tan XDX.
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Now it doesn't look much like
one of the examples. We've just
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been talking about, but we know
that we can redefine Tan X sign
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X over cause X.
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And now when we look at the
derivative of cars is minus sign
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so. The numerator is very nearly
the derivative of the
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denominator, so let's make it
so. Let's put in minus sign X.
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Now having putting the minus
sign, we've achieved what we
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want. The numerator is the
derivative of the denominator,
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the top is the derivative of the
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bottom. But we need to put in
that balancing minus sign so
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that we can retain the
equality of these two
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expressions. Having done
that, we can now write this
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down.
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Minus and it's that minus sign.
The log of caused X plus a
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constant of integration, see.
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We're subtracting a log, which
means we're dividing by what's
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within the log.
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Function, so we're dividing
by cause what we do know is
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with. With dividing by
cars, then that's one over
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cars and that sank. So this
is log of sex X Plus C.
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Now let's go on again and have a
look at another example.
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Integral of X
over one plus
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X squared DX.
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Look at the bottom and
differentiate it. Its derivative
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is 2X only got X on top,
that's no problem. Let's make it
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2X on top by multiplying by two.
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If we've multiplied by two,
we've got to divide by two,
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and that means we want a
half of that result there.
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So now this is balanced out
and it's the same as that.
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What we've got on the top
now is very definitely the
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derivative of what's on
the bottom, so again, we
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can have a half the log of
one plus X squared plus C.
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We can even have this with
look like very complicated
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functions, so let's have a
look at one over X Times
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the natural log of X.
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Doesn't look like what we've got
does it? But let's remember that
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the derivative of log X is one
over X. So if I write this a
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little bit differently, one over
X divided by log X DX.
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Then we can see that what's on
top is indeed the derivative of
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what's on the bottom, and so,
again, this is the log of.
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Log of X plus a constant of
integration. See, so even in
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something like that we can find
what it is we're actually
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looking for. Let's take one more
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example. A little bit contrived,
but it does show you how you
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need to work and look to see.
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If what you've got on the.
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Numerator is in fact the
derivative of the denominator.
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So let's have a look at this.
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Looks quite fearsome as it's
written, but let's just think
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about what we would get if we
differentiate it X Sign X. I'll
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just do that over here. Let's
say Y equals X sign X. Now this
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is a product, it's a U times by
AV, so we know that if Y equals
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UV when we do the
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differentiation. Why by DX
is UDV by the X
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00:06:53,590 --> 00:06:56,790
plus VDU by DX?
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00:06:57,720 --> 00:07:04,908
So in this case you is
X&V is synex, so that's X
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00:07:04,908 --> 00:07:09,101
cause X Plus V which is
synex.
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Times by du by DX, but you was X
or do you buy X is just one?
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So if we look what we can see
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here. Is that the numerator?
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00:07:23,910 --> 00:07:29,562
Is X cause X sign X, which is
the derivative of the
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00:07:29,562 --> 00:07:34,272
denominator X sign X, and so
again complicated though it
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looks we've been able to spot
that the numerator is again
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the derivative of the
denominator, and so we can say
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straight away that the result
of this integral is the log of
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the denominator X Sign X.
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Sometimes you have to look very
closely and let's just remember
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if we just look back at this
one. But sometimes you might
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have to balance the function in
order to be able to make it
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look like you want it to look.
But quite often it's fairly
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clear that that's what you have
to do. So do remember, this is
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a very typical standard form of
integration of very important
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one, and one that occurs a
great deal when looking at
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differential equations.