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
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