Integration of $ \int \frac{2\sin x + \cos x}{\sin x + 2\cos x} dx $ How can I integrate this by changing variable?
$$ \int  \frac{2\sin x + \cos x}{\sin x + 2\cos x} dx $$
Thanks.

Notice that in this case by doing some rearrangement
$$\frac{p(x)}{q(x)}=A\frac{q(x)}{q(x)}+B\frac{q'(x)}{q(x)}$$
$$\begin{align}
\int\frac{2\sin x + \cos x}{\sin x + 2\cos x}dx&=-\frac{3}{5}\int\frac{\cos x-2\sin x}{\sin x + 2\cos x}dx+\frac{4}{5}\int\frac{\sin x+2\cos x}{\sin x + 2\cos x}dx\\
\end{align}$$

$$\int\frac{2\sin x + \cos x}{\sin x + 2\cos x}dx=\frac{1}{5}\left(4x-3\log(\sin x+2\cos x)\right)+C$$

