Integrate $\int_0^{\pi/2}\frac{\cos^2x}{a\cos^2x + b\sin^2x}\,dx$ I don't know how to deal with this integral
$$I=\displaystyle\int_0^{\pi/2}\frac{\cos^2x}{a\cos^2x + b\sin^2x}\,dx$$
I reached the step
$$I
=\displaystyle\ \int_0^{\pi/2}\frac{1}{a + b\tan^2x}dx$$
Now what should I do? Please help.

Continue with
\begin{align}
\int_0^{\pi/2}\frac{1}{a + b\tan^2x}dx
= &\ \frac1{a-b}\int_0^{\pi/2}\bigg(1-  \frac{b\sec^2x}{a + b\tan^2x}\bigg)dx\\
=& \ \frac1{a-b}\bigg( \frac\pi2- \frac\pi2 \sqrt{\frac ba}\bigg)
=\frac\pi{2(a+\sqrt{ab})}
\end{align}
