Calculate integral $\int_{0}^{2\pi} \frac{dx}{a^2\sin^2x+b^2\cos^2x}$

Question

Any hints of how to approach solving this integral?

$$\int_{0}^{2\pi} \frac{dx}{a^2\sin^2x+b^2\cos^2x}, \quad a,b>0$$

I tried substituting $z=e^{ix}$ but I get a rather complex form I can't do anything with.

Answer 1

Try converting the limit to $ 0 $ to $\pi/2$ and then split it into two integral of limits $ 0 $ to $\pi/4$ and $ \pi/4$ to $\pi/2$ . Substitute u=tan x in first integral and v=cot x in second.