Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin x}{\sin x + \cos x}\,dx$.

Explanation

1. 01

   Call the integral $I$. By $\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$, replacing $x$ by $\frac{\pi}{2} - x$ swaps $\sin$ and $\cos$: $I = \int_0^{\frac{\pi}{2}} \frac{\cos x}{\cos x + \sin x}\,dx$.

2. 02

   Adding the two forms, the integrand becomes 1, so $2I = \frac{\pi}{2}$ and $I = \frac{\pi}{4}$.