What is ∫_0^(π/2) (a + sin x)/(2a + sin x + cos x) dx equal to?

Use the property ∫_0^(π/2) f(x) dx = ∫_0^(π/2) f(π/2 - x) dx. Replacing x by π/2 - x: the integrand becomes (a + cos x)/(2a + cos x + sin x). Adding the original and substituted integrals: 2I = ∫_0^(π/2) (2a + sin x + cos x)/(2a + sin x + cos x) dx = ∫_0^(π/2) 1 dx = π/2. Hence I = π/4.