First slide

Evaluation of definite integrals

Question

The value of the integral
$\int_{- 1 / \sqrt{3}}^{1 / \sqrt{3}} \frac{x^{4}}{1 - x^{4}} \cos^{- 1} \frac{2 x}{1 + x^{2}} d x$ is

Moderate

A

0
B

$\frac{π}{\sqrt{3}} + \log \frac{\sqrt{3} + 1}{\sqrt{3} - 1}$
C

$\frac{1}{\sqrt{3}} + \frac{π}{2} \log \frac{\sqrt{3} + 1}{\sqrt{3} - 1}$
D

none of these

Solution

$\int_{- 1 / \sqrt{3}}^{1 / \sqrt{3}} \frac{x^{4}}{1 - x^{4}} \cos^{- 1} \frac{2 x}{1 + x^{2}} d x$
$\begin{array}{l} = \int_{- 1 / \sqrt{3}}^{1 / \sqrt{3}} \frac{x^{4}}{1 - x^{4}} (\frac{π}{2} - \sin^{- 1} \frac{2 x}{1 + x^{2}}) dx \\ = \frac{π}{2} \int_{- 1 / \sqrt{3}}^{1 / \sqrt{3}} \frac{x^{4}}{1 - x^{4}} d x \end{array}$
$(\sin^{- 1} (- x) = - \sin^{- 1} x$ so the last integral is zero)
$\begin{array}{l} = π \int_{0}^{1 / \sqrt{3}} (- 1 + \frac{1}{1 - x^{4}}) d x \\ = π \int_{0}^{1 / \sqrt{3}} (- 1 + \frac{1}{2} (\frac{1}{1 - x^{2}} + \frac{1}{1 + x^{2}})) d x \\ = [- \frac{π}{\sqrt{3}} + {\frac{π}{2} [\frac{1}{2} \log \frac{1 + x}{1 - x} + \tan^{- 1} x]|}_{0}^{1 / \sqrt{3}}] \\ = - \frac{π}{\sqrt{3}} + \frac{π}{2} [\frac{1}{2} \log \frac{\sqrt{3} + 1}{\sqrt{3} - 1} + \frac{π}{6}] \end{array}$

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