The integral ∫ 1 + 2 x( cosecx + x ) dx, (0 < x < (π)/(2)) is equal...

Let $I = ∫ \sqrt{1 + 2\cot x\left(cosecx + \cot x\right)}dx$

$⇒ I = ∫ \left(\sqrt{2\cot x \cos ecx + {cosec}^{2}x + \cot^{2}x}\right)dx$

$⇒ I = ∫ \sqrt{{\left(cosecx + \cot x\right)}^{2}}dx$

$⇒ I = ∫ \left|cosecx + \cot x\right|dx$

$⇒ I = ∫ \left(cosecx + \cot x\right)dx$

$⇒ I = \log \left|\tan\dfrac{x}{2}\right| + \log \left|\sin x\right| + c$, where $c$ is the constant of integration.
$⇒ I = \log \left(\dfrac{sin\dfrac{x}{2}}{cos\dfrac{x}{2}} \cdot 2sin\dfrac{x}{2}cos\dfrac{x}{2}\right) + c$
$⇒ I = 2\log \left|\sin\dfrac{x}{2}\right| + c$