How to do the following integral: ∫01xa−1log(x)dx where a≥0 ?
How to do the following integral:
Answer & Explanation
Call your integral . Then
as long as . Now you need to solve the differential equation
This is a very easy differential equation to solve, and the solution is
where C is some constant. Now we ask, what is that constant? Notice that
so we need
or rather . So we conclude that
as you suggested.
We can utilize
combined with the substitution , to get
We have . Hence, the integral is