First slide

Evaluation of definite integrals

Question

$\int_{0}^{π} \frac{x}{1 + \cos^{2} x} d x =$

Moderate

A

$\frac{1}{2 \sqrt{2}}$
B

$\frac{π^{2}}{2}$
C

$\frac{π^{2}}{\sqrt{2}}$
D

$\frac{π^{2}}{2 \sqrt{2}}$

Solution

$I = \int_{0}^{π} x (\frac{1}{1 + C o s^{2} x}) d x$
     $= \frac{π + 0}{2} \int_{0}^{π} \frac{1}{1 + C o s^{2} x} d x [∵ \int_{a}^{b} x f (x) d x = \frac{b + a}{2} \int_{a}^{b} f (x) d x i f f (a + b - x) = f (x)]$
     $= \frac{π}{2} \int_{0}^{π} \frac{1}{2 C o s^{2} x + S i n^{2} x} d x$
     $= \frac{π}{2} (\frac{π}{a b}) = \frac{π^{2}}{2 \times \sqrt{2} \times 1} = \frac{π^{2}}{2 \sqrt{2}}$

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