$$\int_0^{\frac{\pi}{2}}\frac{a^2}{a^2+\cos^2(x)}\mathrm{d}x$$
The question is really definite integral from $0$ to $\frac{\pi}{2}$. I've tried Wolfram on the indefinite integral: it gives complicated expression.
Is there some trick to calculating definite and if not indefinite then?
Divide all terms by $cos^2x$ and use $sec^2x=tan^2x+1$ to replace the $sec^2x$ in the denominator. Finally u-sub $tanx=t$ You then need to integrate $\frac{a^2dt}{a^2+1+a^2t^2}$ from $t=0$ to $t=inf$ Can you do that? (You will end up with an arctan)