First slide

Evaluation of definite integrals

Question

$\int_{0}^{π / 2} \sin 2 x \log \tan x dx$ is equal to

Moderate

A

$π$
B

$\frac{π}{2}$
C

0
D

$2 π$

Solution

$\begin{array}{l} Let I = \int_{0}^{π / 2} \sin 2 xlog \tan xdx - - - - - (i) \\ \begin{aligned} I & = \int_{0}^{π / 2} \sin 2 (\frac{π}{2} - x) \log \tan (\frac{π}{2} - x) dx \\ [∵ \int_{0}^{a} f (x) dx = \int_{0}^{a} f (a - x) dx] \end{aligned} \\ I = \int_{0}^{π / 2} \sin 2 xlog \cot xdx - - - - (ii) \end{array}$
On adding Eqs. (i) and (ii), we get
$\begin{array}{l} 2 I = \int_{0}^{π / 2} \sin 2 xlog (\tan xcot x) dx \\ = \int_{0}^{π / 2} \sin 2 xlog 1 dx = \int_{0}^{π / 2} 0 dx \\ \Rightarrow I = 0 \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App