Evaluating $\int_{0}^{2\pi } \frac{\sin^{2} (x) }{5+4\cos(x)}\,\mathrm dx$

Question

$$\int_{0}^{2\pi } \frac{\sin^{2} (x) }{5+4\cos(x)}\,\mathrm dx$$

I am having trouble parsing the square of sine in the numerator.

Could someone provide some hint?

Thanks.

Answer 1

use that $$\sin(x)=\frac{2t}{1+t^2}$$ and $$\cos(x)=\frac{1-t^2}{1+t^2}$$ and $$dx=\frac{2}{1+t^2}dt$$ and our indefinite integral is now $$8\int \frac{t^2}{(t^2+9)(t^2+1)^2}dt$$