The value of the integral Ι = ∫ (dx)/(1 + x), x ∈ [0, (π)/(2)] is...

As, $1 + sin x = sin^{2}\dfrac{x}{2} + cos^{2}\dfrac{x}{2} + 2sin\dfrac{x}{2}cos\dfrac{x}{2},$
$= {\left(sin\dfrac{x}{2} + cos\dfrac{x}{2}\right)}^{2}$
$Ι = ∫ \dfrac{dx}{sin\dfrac{x}{2} + cos\dfrac{x}{2}}$
$= \dfrac{1}{\sqrt{2}}∫ \dfrac{dx}{\sin \left(\dfrac{\pi}{4} + \dfrac{x}{2}\right)}$
$= \dfrac{1}{2} ∫ cosec\left(\dfrac{\pi}{4} + \dfrac{x}{2}\right)dx$
$= \dfrac{1}{\sqrt{2}}2\ln \left(\tan \left(\dfrac{\pi}{8} + \dfrac{x}{4}\right)\right) + c$
$= \sqrt{2}\ln \left(\tan \left(\dfrac{\pi}{8} + \dfrac{x}{4}\right)\right) + c$
$⇒ k = \sqrt{2}$

Hence, $\sqrt{2}$k = 2$</p>$