<s>[INST] Evaluate $ \int_{\pi/4}^{\pi/2} \frac{2\sin x+\cos x}{\sin x+2\cos x}\,dx$ 
Evaluate 
  $$ \int_{\pi/4}^{\pi/2} \frac{2\sin x+\cos x}{\sin x+2\cos x}\,dx,$$

My attempt : $u=\tan\frac{x}{2} \rightarrow x=2\arctan(u) \rightarrow \frac{2}{1+u^2}du=dx$
$$\sin x= \frac{2\tan\frac{x}{2}}{1+\tan^2\frac{x}{2}}=\frac{2u}{1+u^2}$$
$$\cos x= \frac{1-\tan^2\frac{x}{2}}{1+\tan^2\frac{x}{2}}= \frac{1-u^2}{1+u^2}$$
I was looking to see if this is the right way to go and also i am not sure how to evaluate the boundaries . Could you help me out ?
 [/INST] Hint:
$$2\sin x+\cos x=\frac{4}{5}(\sin x+2\cos x)+\frac{3}{5}(2\sin x-\cos x)$$
This makes the integral $\displaystyle \int\left(\frac{4}{5}\mathrm dx-\frac{3}{5}\mathrm d(\ln|\sin x+2\cos x|)\right)$ with the same limits. 
</s>