Ans: (D)
Hint \(: N=\int_{0}^{\pi / 4} \frac{\sin 2 x d x}{2(x+1)^{2}}\)
let \(2 x=t, \quad d t=2 d x, \quad N=\int_{0}^{\pi / 2} \frac{\sin t \frac{d t}{2}}{2 \frac{(t+2)^{2}}{4}} \Rightarrow N=\int_{0}^{\pi / 2} \frac{\sin t d t}{(t+2)^{2}}=\left(\frac{-\sin t}{t+2}\right)_{0}^{\pi / 2}+\int_{0}^{\pi / 2} \frac{\cos t}{(t+2)} d t\)
$$
\Rightarrow \mathrm{N}=\frac{-2}{\pi+4}+\mathrm{M} \Rightarrow \mathrm{M}-\mathrm{N}=\frac{2}{\pi+4}
$$
