Evaluate: ∫_0^((π)/2) (sin^2 x)/(1+ sin 2 x) d x

Let, $\displaystyle I=\int_{0}^{\pi / 2} \dfrac{\sin^{2} x}{1+\sin 2x} dx \quad \ldots(1)$

Getting $\displaystyle I=\int_{0}^{\pi / 2} \dfrac{\cos^{2} x}{1+\sin 2x} dx \quad \ldots(2)$

On adding equations ($\displaystyle 1$) and ($\displaystyle 2$), we get

$\displaystyle 2 I=\int_{0}^{\pi / 2} \dfrac{1}{1+\sin 2x} dx$

$\displaystyle \Rightarrow 2 I=\int_{0}^{\pi / 2} \dfrac{1-\sin 2x}{\cos^{2} 2x} dx=\int_{0}^{\pi / 2} \sec^{2} 2x\, dx-\int_{0}^{\pi / 2} \sec 2x \tan 2x\, dx$

$\displaystyle \Rightarrow 2 I=\left[\dfrac{\tan 2x}{2}-\dfrac{\sec 2x}{2}\right]_{0}^{\pi / 2} \Rightarrow I=\dfrac{1}{2}$