Evaluate the integral ∫0π11+2sin2x dx\int_0^{\pi} \frac{1}{1 + 2\sin^2 x} \, dx∫0π1+2sin2x1dx.
π3\frac{\pi}{\sqrt{3}}3π
π2\frac{\pi}{\sqrt{2}}2π
π\piπ
π23\frac{\pi}{2\sqrt{3}}23π