Mathematics

# Solve  $\displaystyle \int { \dfrac { \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+2 \right) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }+4 \right) } } dx$

##### SOLUTION
$\int{\dfrac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}}dx$

This can be written as

$\Rightarrow \int{(\dfrac{-4x^2-10}{(x^2+3)(x^2+4)}}+1)dx$

$\Rightarrow \int{\dfrac{-4x^2-10}{(x^2+3)(x^2+4)}}dx+\int 1dx$

$\Rightarrow -2\int{\dfrac{2x^2+5}{(x^2+3)(x^2+4)}}dx+\int 1dx$

$\Rightarrow -2\int{\dfrac{3}{x^2+4}}dx+2\int{\dfrac{1}{x^2+3}}dx+x$

$\Rightarrow -6\int{\dfrac{1}{x^2+4}}dx+2\int{\dfrac{1}{x^2+3}}dx+x$

$\Rightarrow -\dfrac{6}{2}\tan ^{-1}(\dfrac{x}{2})+\dfrac{2}{\sqrt 3}\tan ^{-1}(\dfrac{x}{\sqrt 3})+x$

$\Rightarrow x -3\tan ^{-1}(\dfrac{x}{2})+\dfrac{2}{\sqrt 3}\tan ^{-1}(\dfrac{x}{\sqrt 3})$

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

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