_ / 4 ^ / 2 e ^x( x+ x) d x= - Vidyadip

MCQ

$\int_{\pi / 4}^{\pi / 2} e ^x(\log \sin x+\cot x) d x=$

A
$e ^{\frac{\pi}{4}} \log 2$
B
$- e ^{\frac{\pi}{4}} \log 2$
✓
$\frac{1}{2} e ^{\frac{\pi}{4}} \log 2$
D
$-\frac{1}{2} e ^{\frac{\pi}{4}} \log 2$

✓

Answer

Correct option: C.

$\frac{1}{2} e ^{\frac{\pi}{4}} \log 2$

(C)
$\int_{\pi / 4}^{\pi / 2} e ^x(\log \sin x+\cot x) d x$
$\begin{array}{l}=\left[ e ^x \log \sin x\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} \\ = e ^{\frac{\pi}{2}} \log \sin \frac{\pi}{2}- e ^{\frac{\pi}{4}} \log \sin \frac{\pi}{4}=\frac{1}{2} e ^{\frac{\pi}{4}} \log 2\end{array}$

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