use the substitution x = e^t and then perform integration by parts.
we could split the integral up and find,
the last integral can be done using the power rule, for the first, i see two ways of approaching it.
the easier of the two ways is to make the substitution , and continue by integration by parts, which will be easy (you may even know the integral of ln x by heart, so integration by parts won't be necessary)
the second way is to multiply the first integral by and then use the substitution . again you would continue by parts, but this one would be a little harder i think
for the benefit of the class, i shall show how we find the integral of ln(x) using the by parts method.
the trick here is to note that , obvious enough
Thus, if we take and , we have:
An alternative method to find the integral of is the following. (i actually just thought of this now, i've never seen this done before, but the technique is a very familiar one)
By substitution, let
Thus, our integral becomes
of course, this way is a bit more work than the last, but it's nice to know several ways to skin a cat...if skinning a cat is your thing