Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Explain solution RD Sharma class 12 chapter 19 Definite Integrals exercise Multiple choice question 37 maths

Explain solution RD Sharma class 12 chapter 19 Definite Integrals exercise Multiple choice question 37 maths

Answers (1)

Answer:

\pi \log 2

Hint:

To solve this equation we suppose x as tan x.

Given:

\int_{0}^{\infty} \log \left(x+\frac{1}{x}\right)\left(\frac{1}{1+x^{2}}\right) d x

Solution:

Let

\begin{aligned} &I=\int_{0}^{\infty} \log \left(x+\frac{1}{x}\right)\left(\frac{1}{1+x^{2}}\right) d x \\\\ &x=\tan x \end{aligned}

\begin{aligned} &d x=\sec ^{2} \theta d \theta \\\\ &\text { When } x=0, \tan \theta=0, \theta=0 \\\\ &\text { When } x=\infty, \tan \theta=\infty, \theta=\frac{\pi}{2} \end{aligned}

\begin{aligned} &I=\int_{0}^{\frac{\pi}{2}} \log (\tan \theta+\cot \theta) \cdot \frac{\sec ^{2} \theta d \theta}{1+\tan ^{2} \theta} \\\\ &=\int_{0}^{\frac{\pi}{2}} \log \left(\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}\right) d \theta \end{aligned}

\begin{aligned} &=\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta} d \theta \\\\ &=\int_{0}^{\frac{\pi}{2}} \log 1-\log (\sin \theta \cos \theta) d \theta \end{aligned}

\begin{aligned} &=\int_{0}^{\frac{\pi}{2}} \log \sin \theta d \theta-\int_{0}^{\frac{\pi}{2}} \log \cos \theta d \theta \\\\ &=-\left(\frac{\pi}{2} \log 2\right)-\left(-\frac{\pi}{2} \log 2\right) \end{aligned}

I=0

Posted by

infoexpert26

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE admits Class 12 answer sheet error; assures revised result for affected students
anam-khan

‘Compare and see difference’: Students claim CBSE 12th answer sheet mismatch; allege marks discrepancy
anam-khan

CBSE payment gateway overhaul: Govt orders major fix after transaction glitches

Latest Question