Integral: $\mathrm{{\int_{0}^{\pi}{\frac{3\cdot cos(x)\cdot sin(x)+4\cdot sin(x)}{sin^{2}(x)+2\cdot cos^{2}(x)-cos(x)}\, dx}}}$

Question

We are seeking to evaluate this trigonometric and fraction integral in $\mathrm{sin}$ and $\mathrm{cos}$:

$$\mathrm{{\int_{0}^{\pi}{\frac{3\cdot cos(x)\cdot sin(x)+4\cdot sin(x)}{sin^{2}(x)+2\cdot cos^{2}(x)-cos(x)}\, dx}}}$$

I tried with $\mathrm{sin^{2}(x)+cos^{2}(x)}$, but I can't go on.

Thanks.

Answer 1

Hint. Let $t=\cos(x)$, then $$\int_{0}^{\pi}{\frac{3\cos(x)\sin(x)+4\sin(x)}{\sin^{2}(x)+2\cos^{2}(x)-\cos(x)}\, dx}=\int_{-1}^{1}{\frac{3t+4}{1+t^2-t}\, dt}.$$ It remains to integrate a rational function. Can you take it from here?