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We are faced with a fairly daunting looking
indefinite integral of pi over x natural log of x dx.
Now what could we do to address this?
Is u-substitution a possibility here?
Well, for u substitution, we want to look for an expression and its derivative.
Well, what happens if we set u equal to the natural log of x?
Now what would "d u" be equal to in that scenario?
"du" is going to be the derivative of the natural log of x with respect to x
which is just 1 over x d x.
This is an equivalent statement to saying that d u d x is equal to one over x.
So do we see the one over x anywhere in this original expression?
Well, it's kind of hiding
it's not so obvious, but this x in the denominator is essentially
one over x and that's being multiplied by dx.
Let me rewrite this original expression to make a little bit more sense.
So the first thing I'm gonna do, is I'm gonna take
the pi - I should do that in a different color
since I've already used...
let me take the pi and just take it out front.
Imma just take the pi right in front of the integral.
And so this is gonna become the integral of -
and let me write the one over the natural log of x first.
one over the natural log of x
times one over x
dx
Now it becomes a little bit clearer.
These are completely equivalent statements.
But this makes it clear that yes,
u substitution will work over here.
We set our u equal to natural log of x.
Then our d u is one over x d x.
One over x d x.
Our d u is one over x d x.
Let's rewrite this integral.
It's going to be equal to
pi times the indefinite integral of
one over u
natural log of x is u
we set that equal to natural log of x
times du
times du
Now this becomes pretty straightforward.
What is the antiderivative of all this business?
And we've done very
similar things like this multiple times already.
This is going to be equal to
pi
times the natural log
the natural log of the absolute value
of u
so that we can handle even negative values of u
the natural log of the absolute value of u
plus c
ok, so we have a constant factor out here.
plus c
And we're almost done, we just have to un-substitute for the u.
U is equal to natural log of x.
So we end up with this kind of neat looking expression.
The anti of this entire indefinite integral, we have simplified
we have evaluated it
and it is now equal to pi
times the natural log
of the absolute value of u
but u is just the natural log of x
the natural log of x
and then we have this plus c
right over here.
And we could've assumed that from the get-go
this original expression
was only defined for positive values of x
because you have to take the natural log here
and there wasn't a absolute value.
So we can leave this as just a natural log of x
But this also works for the situations now
cause we're taking the absolute value of that
where the ln of x might have been a negative number.
for example if it was a natural log of
.5 or who knows whoever it might be.
But we are all done.
We have simplified what seemed
like a kind of daunting expression.