Mathematics

# $\int_{0}^{\infty }\frac{log(1+x^{2})}{1+x^{2}}dx$

##### SOLUTION

Consider the given function:

$f(x)=\int_{0}^{\infty }{\dfrac{\log (1+{{x}^{2}})}{(1+{{x}^{2}})}}dx$

$\,\,\,\,\,\,\,\,\,\,\,\,let\,\,x=\sec \theta$

$=\int_{0}^{\dfrac{\pi }{2}}{{{\sec }^{2}}\theta \dfrac{\log ({{\sec }^{2}}\theta )}{{{\sec }^{2}}\theta }}\,\,d\theta$

$=\int_{0}^{\dfrac{\pi }{2}}{2\log {{\sec }^{2}}\theta }\,\,d\theta$

$=\int_{0}^{\dfrac{\pi }{2}}{-2\log \cos \theta }\,\,d\theta$

$=-2*\int_{0}^{\dfrac{\pi }{2}}{\log \cos \theta \,\,d\theta }$

$=-2*(-\dfrac{\pi }{2})\log 2$

$=\pi \log 2$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Subjective Medium
Integrate :
$\dfrac {{e}^{2x}-1}{{e}^{2x}+1}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Multiple Correct Hard
If $I_{1}= \displaystyle \int ^{1}_{x}\displaystyle \frac{dt}{1+t^{2}}$ and $I_{2}= \displaystyle \int ^{1/x}_{1}\displaystyle \frac{dt}{1+t^{2}}$ for $x> 0$, then
• A. $I_{1}> I_{2}$
• B. $I_{2}> I_{1}$
• C. $I_{1}= I_{2}$
• D. $I_{2}= \pi /4-\tan ^{-1}x$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int\frac{\sin 2x}{a^{2}+b^{2}\sin^{2}x}dx=$
• A. $\dfrac{1}{a^{2}-b^{2}}\log|a^{2}+b^{2}\sin^{2}x|+c$
• B. $\dfrac{1}{a^{2}}\log|a^{2}+b^{2}\sin^{2}x|+c$
• C. $\displaystyle \frac{1}{a^{2}+b^{2}}\log|a^{2}+b^{2}\sin^{2}x|+c$
• D. $\displaystyle \frac{1}{b^{2}}\log|a^{2}+b^{2}\sin^{2}x|+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 One Word Hard
If ${I}_{1}=\displaystyle\int_{0}^{1}{\dfrac{{\tan}^{-1}{x}\,dx}{x}}$ and ${I}_{2}=\displaystyle\int_{0}^{\frac{\pi}{2}}{\dfrac{x\,dx}{\sin{x}}}$ then $\dfrac{{I}_{1}}{{I}_{2}}$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$