Question

$\int e^{-x}\text{cosec}x(\cot x+1)dx=$

• A. $-e^{-x}\text{cosec}x+c$
• B. $e^x\text{cosec}x+c$
• C. $e^x\cot x+c$
• D. $-e^{-x}\cot x+c$

Answer 1

Answer: (A)

$-e^{-x}\text{cosec}x+c$

$I=\int e^{-x}\text{cosec}x(\cot x+1)dx$
Substitute, $x=-t\Rightarrow dx=-dt$
$-\int e^t\text{cosec}(-t)\left(\cot \right(-t)+1)dt$
$\Rightarrow \int e^t\text{cosec}t(-\cot t+1)dt$
$\Rightarrow \int e^t(\text{cosec}t-\text{cosec}t\cot t)dt$
It is in the form of
$\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=e^{x}f\left(x\right)+c$
$\therefore I=e^t\text{cosec}t+c$
$I=e^{-x}\text{cosec}(-x)+c$
$I=-e^{-x}\text{cosec}x+c$