Question

The integral $\int \sqrt{\cot x}{e}^{\sqrt{\sin x}}\sqrt{\cos x}dx$ equals

Answer 1

$\int \sqrt{\cot x}{e}^{\sqrt{\sin x}}\sqrt{\cos x}dx$

$=\int \sqrt{\cot x\cos x}{e}^{\sqrt{\sin x}}dx$

$=\int \sqrt{\frac{\cos x}{\sin x}\cos x}{e}^{\sqrt{\sin x}}dx$

$=\int \frac{\cos x}{\sqrt{\sin x}}{e}^{\sqrt{\sin x}}dx$

Let $t=\sqrt{\sin x}\Rightarrow dt=\frac{\cos x}{2\sqrt{\sin x}}dx$

$=\int 2e^{t}dt$

$=2\int e^{t}dt$

$=2e^t+c$

$=2e^{\sqrt{\sin x}}+c$ where $t=\sqrt{\sin x}$