Mathematics

# Evaluate: $\int { \cfrac { x+3 }{ \left( x-1 \right) \left( { x }^{ 2 }+1 \right) } } dx$

##### SOLUTION
Let $\dfrac{x+3}{(x-1)(x^{2}+1)}=\dfrac{A}{x-1}+\dfrac{B{x}+C}{x^{2}+1}$
$\implies A(x^{2}+1)+(B{x}+C)(x-1)=x+3$
on solving this we get $A=2,B=-2,C=-1$
$\displaystyle\int \dfrac{x+3}{(x-3)(x^{2}+1)}dx=2\displaystyle\int \dfrac{dx}{x-1}-\displaystyle\int \dfrac{2{x}{d{x}}}{x^{2}+1}-\displaystyle\int \dfrac{d{x}}{x^{2}+1}=2\ln(x-1)-\ln (x^{2}+1)-\text{tan}^{-1}x+C$

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

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