$\int e^{x \operatorname{cosec} x} \cdot \operatorname{cosec} x \cdot(1-x \cot x) d x=$

Question

Aicharya · Accepted Answer

$\int e^{x \operatorname{cosec}} x \operatorname{cosec} x(1-x \cot x) d x$ Put, $x \operatorname{cosec} x=t$ Differentiate w.r.to ' $x$ ' $ \begin{aligned} & x \cdot \frac{d}{d x} \operatorname{cosec} x+\operatorname{cosec} x \cdot \frac{d}{d x} \cdot x=\frac{d}{d x} t \ & x(-\operatorname{cosec} x \cdot \cot x)+\operatorname{cosec} x(1)=\frac{d t}{d x} \ & \operatorname{cosec} x(1-x \cot x) d x=d t=\int e^t \cdot d t \ &=e^t+c=e^{x \operatorname{cosec} x}+c \end{aligned} $ Hence, option (2) is correct

$\int e^{x \operatorname{cosec} x} \cdot \operatorname{cosec} x \cdot(1-x \cot x) d x=$

Solution