Evaluate: ∫₀^(π/2) (cos² x)/(1 + sin x cos x) dx.

Let $I = \int_0^{\pi/2}\frac{\cos^2x}{1+\sin x\cos x}dx$. By $\int_0^a f(x)dx = \int_0^a f(a-x)dx$: $I = \int_0^{\pi/2}\frac{\sin^2x}{1+\sin x\cos x}dx$. Adding: $2I = \int_0^{\pi/2}\frac{dx}{1+\sin x\cos x}$. Divide numerator and denominator by $\cos^2x$ and put $\tan x = t$, $\sec^2x\,dx = dt$: $2I = \int_0^\infty\frac{dt}{t^2+t+1} = \int_0^\infty\frac{dt}{\left(t+\frac12\right)^2+\frac34}$ $= \frac{2}{\sqrt3}\left[\tan^{-1}\frac{2t+1}{\sqrt3}\right]_0^\infty = \frac{2}{\sqrt3}\left(\frac\pi2 - \frac\pi6\right) = \frac{2\pi}{3\sqrt3}$ Hence $I = \frac{\pi}{3\sqrt3} = \frac{\sqrt3\,\pi}{9}$.