Help with solving 2 integrals - one definite and one indefinite

Question

I need help with solving these two integrals:

$$\begin{eqnarray*} &\int_{ -\pi /4}^{ \pi /4} \frac{x^7-x+1}{\cos^2 x}\mathrm dx \\ &\int \frac{\sin^2(x)}{\sin(x)+2\cos(x)}\mathrm dx \end{eqnarray*}$$

Answer 1

For the Second one @Dr Sonnhard already give a sufficient hint.

For the First one observe that

$$\int_{ -\pi /4}^{ \pi /4} \frac{x^7-x+1}{\cos^2 x}\mathrm dx =\int_{ -\pi /4}^{ \pi /4} \frac{1}{\cos^2 x}\mathrm dx =\int_{ -\pi /4}^{ \pi /4} (\tan x)'dx =2 $$

Given that

$$\int_{ -\pi /4}^{ \pi /4} \frac{x^7-x}{\cos^2 x}\mathrm dx =0$$ since the integral is over a symetric domain and the integrand is an odd function.

also note that $$ (\tan x)'= 1+\tan ^2x =\frac{1}{\cos^2 x}$$