Mathematics

# Evaluate $\displaystyle \int { \cfrac { { x }^{ 3 }+1 }{ { x }^{ 2 }+1 } } dx=$

##### SOLUTION
$\displaystyle\int \dfrac{x^{3}+1}{x^{2}+1}dx=\int \dfrac{(x+1)(x^{2}+1-x)}{(x^{2}+1)}$
$=\displaystyle\int (1-\dfrac{x}{x^{2}+1})(x+1)$
$=\displaystyle\int (x+1)dx-\int \dfrac{x^{2}+x}{x^{2}+1}dx$
$=\displaystyle\int (x+1)dx-\int dx-\int \dfrac{x-1}{x^{2}+1}dx$
$=\displaystyle\int x.dx-\frac{1}{2}\int \dfrac{2x}{x^{2}+1}dx+\int \dfrac{1}{1+x^{2}}dx$
$=\dfrac{x^{2}}{2}-\dfrac{1}{2}dx(1+x^{2})+\tan^{-1}x+c$

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

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