Question

$\int e^x(\log \sin x+\cot x)dx=\_\_\_\_\_\_+C.$

• A. $e^x\cot x$
• B. $e^x\log \sin x$
• C. $e^x\tan x$
• D. None of these

Answer 1

Answer: (B)

$e^x\log \sin x$

$\int e^x\left(\log \sin x+\cot x\right)dx$

As we know

$\begin{array}{c}\int e^x\left[f\left(x\right)+f^{\prime }\left(x\right)\right]dx=e^{x}f\left(x\right)+c\\ f\left(x\right)=\log \sin x\\ f^{\prime }\left(x\right)=\frac{1}{\sin x}.\cos x=\cot x\\ \int e^x\left(\log \sin x+\cot x\right)dx.=e^x\left(\log \sin x\right)+c\end{array}$

Hence, this is the answer.