Question

Evaluate:

$\int e^{xcosecx}\left[\frac{\sin x-x\cos x}{{\sin }^{2}x}\right]dx$

• A. $e^x.cosecx+C$
• B. $e^{xcosecx}+C$
• C. $-e^{xcosecx}+C$
• D. $e^{x\sin }+C$

Answer 1

The correct option is B $e^{xcosecx}+C$

$\int e^{xcosecx}\left[\frac{\sin x-x\cos x}{{\sin }^{2}x}\right]dx$

$=\int e^{xcosecx}[cosecx-x.cosecx.\cot x]dx$

$=\int d\left\{e^{xcosecx}\right\}$ [$\because \int e^{f\left(x\right)}.f^{\prime }\left(x\right)dx=e^{f\left(x\right)}$]

$=e^{xcosecx}+c$ [ Where $c$ being integrating costant]