Mathematics

Find $\displaystyle \int \dfrac{x^4+1}{x(x^2+1)^2}$

SOLUTION
Given $\int{\dfrac{x^4+1}{x(x^2+1)^2}dx}$
$=\int{\left(\dfrac{1}{x}-\dfrac{2x}{(x^2+1)^2}\right)dx}$
$=\int{\dfrac{1}{x}dx}-2\int{\dfrac{x}{(x^2+1)^2}dx}$
$\int{\dfrac{1}{x}dx}=ln(x)$
$\int{\dfrac{x}{(x^2+1)^2dx}}$
Substituting $u=x^2+1$    $dx=\dfrac{1}{2x}du$
$=\dfrac{1}{2}\int{\dfrac{1}{u^2}du}$
$=-\dfrac{1}{2u}$
$=-\dfrac{1}{2(x^2+1)}$
Putting in solved integrals
$=\int{\dfrac{1}{x}dx}-2\int{\dfrac{x}{(x^2+1)^2}dx}$
$=ln(x)+\dfrac{1}{x^2+1}+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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