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  • 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

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