I'm working through an integral suggested for practice at the end of the Wikipedia article on differentiation under the integral sign, and I'm stuck.
I am attempting to evaluate this integral:
$\int_0^{\pi/2} \frac{x}{\tan x} \ dx.$
The article suggests the following parameterization:
$F(a)=\int_0^{\pi/2} \frac{\tan^{-1}(a\tan (x))}{\tan x} \ dx.$
Differentiating with respect to $a$, we get
$F'(a)=\int_0^{\pi/2} \frac{1}{1+a^2\tan^2 x} \ dx.$
I can't find a way to evaluate this, and neither can Wolfram Alpha. The special values $a=0,1$ are easy, but I fail to see how they help.
How can I finish evaluating this integral?
Edit: I think it's just substitution, maybe. I'll update the post accordingly soon.
Edit 2: Indeed, the substitution $u=\tan x$ and identity $\sec^2 = 1 + \tan^2$ transform the above integral into
$F'(a) = \int_0^{\pi/2} \frac{1}{(1+u^2)(1+a^2u^2)}.$
This can be solved with partial fractions.