The integral ∫ 0 π 2 1 3 + 2 sin x + cos x d x is equal to:

Given,

$I={\int }_0^{\frac{\pi }{2}}\frac{dx}{3+2\sin x+\cos x}$

$\Rightarrow I={\int }_0^{\frac{\pi }{2}}\frac{dx}{3+2{\displaystyle \left(\frac{2\tan {\displaystyle \frac{x}{2}}}{1+{\tan }^2{\displaystyle \frac{x}{2}}}\right)}+\frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}}$

$\Rightarrow I={\int }_0^{\frac{\pi }{2}}\frac{{\sec }^2\frac{x}{2}·dx}{2{\tan }^2\frac{x}{2}+4\tan \frac{x}{2}+4}$

Put $\tan \frac{x}{2}=t \Rightarrow \frac{1}{2}{\text{sec}}^2\frac{x}{2}dx=dt$, so

$\Rightarrow I={\int }_0^1\frac{dt}{{\left(t+1\right)}^2+1}$

$\Rightarrow I={\tan }^{-1}{\overline{)\left(t+1\right)}}_0^1$

$\Rightarrow I={\tan }^{-1}2-\frac{\pi }{4}$