A definite integral: $\int_{0}^{\pi/2} \frac{\sin x~\mathrm dx}{\sin x+\cos x+ e^x}$ Mathematica can do this integral,
$$\int_{0}^{\pi/2} \frac{\sin x~ \mathrm dx}{\sin x+\cos x+ e^x}\,,$$
the question is: how to do it by hand?

$$\int_{0}^{\pi/2} \frac{\sin x~ dx}{\sin x+\cos x+ e^x}dx$$
$$=\int_{0}^{\pi/2} \frac{e^{-x}\sin x~ dx}{e^{-x}(\sin x+\cos x)+ 1}dx$$
Put $1+e^{-x}(\sin x+\cos x)=t$. Then,  $-2e^{-x}\sin x dx=dt$.
The integral changes to
$$=\int_{2}^{1+e^{-\pi/2}} \frac{-1}{2t}dt$$
$$=\frac{1}{2}\ln\left(\frac{2}{1+e^{-\pi/2}}\right)$$
