Evaluate the following intregals: x (x+1)(x^2+1) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{\text{x}}{(\text{x}+1)(\text{x}^2+1)}\ \text{dx}$

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Answer

let $\frac{\text{x}}{(\text{x}+1)(\text{x}^2+1)}=\frac{\text{A}}{\text{x}+1}+\frac{\text{Bx}+\text{C}}{\text{x}^2+1}$$\Rightarrow\text{x}=\text{A}(\text{x}^2+1)+(\text{Bx}+\text{C})(\text{x}+1)$
Equation similar terms, we get
$\text{A}+\text{B}=0,\text{B}+\text{C}=1,\text{A}+\text{C}=0$
Solving, we get, $\text{A}=-\frac{1}{2},\text{B}=\frac{1}{2},\text{C}=\frac{1}{2}$
Thus,
$\text{I}=-\frac{1}{2}\int\frac{\text{dx}}{\text{x}+1}+\frac{1}{2}\int\frac{\text{x dx}}{\text{x}^2+1}+\frac{1}{2}\int\frac{\text{dx}}{\text{x}^2+1}$
$\text{I}=-\frac{1}{2}\log|\text{x}+1|+\frac{1}{4}\log|\text{x}^2+1|+\frac{1}{2}\tan^{-1}\text{x}+\text{c}$

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