The integral ∫ _ 0 ^ (π)/(2) (1)/(3 + 2 x + x)dx is equal to:

Given,

$I = {∫}_{0}^{\dfrac{\pi}{2}}\dfrac{dx}{3 + 2\sin x + \cos x}$

$⇒ I = {∫}_{0}^{\dfrac{\pi}{2}}\dfrac{dx}{3 + 2\left(\dfrac{2tan\dfrac{x}{2}}{1 + \tan^{2}\dfrac{x}{2}}\right) + \dfrac{1 - \tan^{2}\dfrac{x}{2}}{1 + \tan^{2}\dfrac{x}{2}}}$

$⇒ I = {∫}_{0}^{\dfrac{\pi}{2}}\dfrac{\sec^{2}\dfrac{x}{2} \cdot dx}{2\tan^{2}\dfrac{x}{2} + 4tan\dfrac{x}{2} + 4}$

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

$⇒ I = {∫}_{0}^{1}\dfrac{dt}{{\left(t + 1\right)}^{2} + 1}$

$⇒ I = \tan^{- 1}{\left(t + 1\right)}_{0}^{1}$

$⇒ I = \tan^{- 1}2 - \dfrac{\pi}{4}$