Explanation

(c)
The primitive of the given function is
$\int_{}^{}{\frac{dx}{\sqrt{\sin^{3}{x\sin{(x + \theta)}}}} = \int_{}^{}\frac{dx}{\sqrt{\sin^{3}{x(\sin{x\cos{\theta + \cos{x\sin{\theta)}}}}}}}}$
$= \int_{}^{}\frac{cosec^{2}x\ dx}{\sqrt{\cos{\theta + \cot{x\sin\theta}}}}$
Put ${\cot{x = t\ \Rightarrow - cosec}}^{2}x\ dx = dt$
$= - \frac{1}{\sqrt{\sin\theta}}\int_{}^{}\frac{dt}{\sqrt{\cot{\theta + 1}}}$
$= - \frac{2}{\sqrt{\sin\theta}}({\cot{\theta + t)}}^{1/2} + c$
$= - \frac{2}{\sin\theta}({\cos{\theta + t\sin{\theta)}}}^{1/2} + c$
$= \frac{- 2cosec\ \theta(\sin{x\cos{\theta + {\sin{\theta\cos{x)}}}^{1/2}}}}{\sqrt{\sin x}} + c$
$= - 2\ cosec\ \theta\left( \ \frac{\sin{(\theta + x)}}{\sin x}\ \right)^{1/2} + c$