Question

$\int \frac{1}{\sqrt{{\sin }^{3}x\sin (x+a)}}dx$ is equal to

• A. $2\cos ec\alpha \sqrt{\cos \alpha +\sin \alpha \tan x}+c$
• B. $-2\cos ec\alpha \sqrt{\cos \alpha +\sin \alpha \cot x}+c$
• C. $\cos ec\alpha \sqrt{\cos \alpha +\sin \alpha \tan x}+c$
• D. $-\cos ec\alpha \sqrt{\cos \alpha +\sin \alpha \tan x}+c$

Answer 1

The correct option is B $-2\cos ec\alpha \sqrt{\cos \alpha +\sin \alpha \cot x}+c$

$\begin{array}{c}\int \frac{dx}{\sqrt{{\sin }^{3}x\sin \left(x+a\right)}}\\ =\int \frac{dx}{\sqrt{{\sin }^{3}x\sin x\left(\cos a+\cot x\sin a\right)}}\\ =\int \frac{\cos e{c}^{2}xdx}{\sqrt{\cos a+\sin a\cot x}}\\ \cos a+\sin a\cot =t\\ \frac{dt}{dx}=-\sin a\cos e{c}^{2}x\\ -\frac{1}{\sin a}dt=\cos e{c}^{2}xdx\\ =-\frac{1}{\sin a}\int \frac{dt}{\sqrt{t}}\\ =-\frac{1}{\sin a}\int t^{\frac{-1}{2}}dt\\ =\frac{-2}{\sin a}{t}^{\frac{1}{2}}+C\\ =-2\cos eca\sqrt{\cos a+\sin a\cot x}+C\end{array}$