Mathematics

# Evaluate the following integral $\int _{ 0 }^{ \infty }{ \dfrac { dx }{ \left( { x }^{ 2 }+{ a }^{ 2 } \right) \left( { x }^{ 2 }+{ b }^{ 2 } \right) } } =$

$\dfrac {\pi}{2ab(a+b)}$

##### SOLUTION
$\displaystyle \int_{0}^{\infty }\frac{dx}{(x^{2}+a^{2})(x^{2}+b^{2})}$
$\displaystyle\Rightarrow \frac{1}{(a^{2}-b^{2})} \int_{0}^{\infty} \frac{(a^{2}-b^{2})dx}{(x^{2}+a^{2})(x^{2}+b^{2})}$
$\displaystyle \Rightarrow \frac{1}{(a^{2}-b^{2})} \int_{0}^{\infty}\left [ \frac{(x^{2}+a^{2})-(x^{2}+b^{2})}{(x^{2}+a^{2})(x^{2}+b^{2})} \right ]dx$
$\displaystyle = \frac{1}{(a^{2}-b^{2})} \left [ \int_{0}^{\infty} \frac{dx}{(x^{2}+b^{2})} -\int_{0}^{\infty} \dfrac{dx}{(x^{2}+a^{2})}\right ]$
$\displaystyle = \dfrac{1}{(a^{2}-b^{2})} \left [ \frac{1}{b} tan^{-1}\frac{x}{b}-\frac{1}{a} tan^{-1}\frac{x}{a} \right ]_{0}^{\infty }$
$\displaystyle = \frac{1}{(a^{2}-b^{2})} \left [ \frac{1}{b}tan^{-1}\infty - \frac{1}{a}tan^{-1} \infty ] -(0-0) \right ]$
$\displaystyle = \dfrac{1}{(a^{2}-b^{2})} \left [ \dfrac{1}{b} \times \dfrac{\pi}{2}-\dfrac{1}{a} \times \dfrac{\pi}{2} \right ]$
$= \dfrac{\pi}{2(a-b)(a+b)} \left [ \dfrac{(a-b)}{ab} \right ]$
$= \dfrac{\pi}{2ab (a+b)}$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium
Evaluate $\displaystyle\int^8_1\dfrac{dx}{x^{2/3}}$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\int { \cfrac { 2{ x }^{ 3 } }{ \left( 4+{ x }^{ 8 } \right) } } dx=$?
• A. $\cfrac { 1 }{ 2 } \tan ^{ -1 }{ \cfrac { { x }^{ 4 } }{ 2 } } +C$
• B. $\cfrac { 1 }{ 2 } \tan ^{ -1 }{ { x }^{ 4 } } +C\quad$
• C. none of these
• D. $\cfrac { 1 }{ 4 } \tan ^{ -1 }{ \cfrac { { x }^{ 4 } }{ 2 } } +C\quad$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\displaystyle \int { \frac { x+\sqrt [ 3 ]{ { x }^{ 2 } } +\sqrt [ 6 ]{ x } }{ x\left( 1+\sqrt [ 3 ]{ x } \right) } dx }$ is equal to
• A. $\displaystyle \frac { 3 }{ 2 } { x }^{ 2/3 }-6\tan ^{ -1 }{ { x }^{ 1/6 } } +c$
• B. $\displaystyle -\frac { 3 }{ 2 } { x }^{ 2/3 }+6\tan ^{ -1 }{ { x }^{ 1/6 } } +c$
• C. None of these
• D. $\displaystyle \frac { 3 }{ 2 } { x }^{ 2/3 }+6\tan ^{ -1 }{ { x }^{ 1/6 } } +c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve:
$\int \dfrac{dx}{\sin^2x\cos^2x}$

Evaluate $\int \dfrac{e^x-e^{-x}}{e^x+e^{-x}}dx$