_0^ / 2 2 x x d x is equal to - Vidyadip

MCQ

$\int_0^{\pi / 2} \sin 2 x \log \tan x d x$ is equal to

A
$\pi$
B
$\frac{\pi}{2}$
✓
$0$
D
$2 \pi$

✓

Answer

Correct option: C.

$0$

(C)
Let $I =\int_0^{\pi / 2} \sin 2 x \log \tan x d x$
$=\int_0^{\pi / 2} \sin 2\left(\frac{\pi}{2}-x\right) \log \tan \left(\frac{\pi}{2}-x\right) d x$
$\ldots\left[\because \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$
$\begin{array}{l}=\int_0^{\pi / 2} \sin 2 x \log \cot x d x \\ =-\int_0^{\pi / 2} \sin 2 x \log \tan x d x\end{array}$
$\therefore \quad I =- I \Rightarrow 2 I =0 \Rightarrow I =0$

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