Evaluate: ∫₀^(π/2) (a sin x + b cos x)/(sin x + cos x) dx

Let $I = \int_0^{\pi/2} \frac{a\sin x + b\cos x}{\sin x + \cos x}\, dx$. Using $\int_0^{c} f(x)\, dx = \int_0^{c} f(c - x)\, dx$: $I = \int_0^{\pi/2} \frac{a\cos x + b\sin x}{\sin x + \cos x}\, dx$ Adding: $2I = (a + b)\int_0^{\pi/2} \frac{\sin x + \cos x}{\sin x + \cos x}\, dx = (a + b)\frac{\pi}{2}$ Hence $I = \frac{(a+b)\pi}{4}$.