TITLE: Integral involving logarithm: $\int_0^\infty \frac{ \ln x}{(x+a)(x+b)} dx$
QUESTION [13 upvotes]: How to solve the following integral
$$\int_{0}^{\infty} \frac{ \ln x}{(x+a)(x+b)} dx,$$
where $a,b>0$ and $a \neq b$. I was looking for some kind of substitution. However, I don't see an obvious one.
Thanks!

REPLY [6 votes]: One way to integrate the $\log$ function is as follows: by parts let $u = \log x$ and $dv = 1$. Then:
$\int u \frac{dv}{dx} = uv - \int v\frac{du}{dx} = x\log x - \int \frac{x}{x} = x\log x - x$
Knowing this and that your integral is a product of three separate integrals you can then repeatedly apply integration by parts twice to the remaining terms.