By differentiating with respect to $\alpha$, find the values of the following integrals: $$\int_0^\pi \ln(1+\alpha\cos x)dx\quad\text{and}\quad \int_0^1\frac{x^\alpha-1}{\ln x}dx.$$
(This is my first question on the site so forgive me if I make any mistakes)
I've been trying to solve the first integral in the attachment for a while now, my problem is the definite integral of the derivative $ \cos x/ (1 + a \cos x)$, where "a" is the parameter, from $0$ to $pi$. I gave up yesterday and checked what the general solution was on Wolfram Alpha and it was definitely beyond what I was capable of. However, the second question was very easy, so I'm wondering if there was a trick involving the limits I missed out on for the first integral that would make it doable.
