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

Question

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

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Answer

Let $\text{x}^2=\text{y}$
Then, $\frac{1}{(\text{y}+1)(\text{y}+2)}=\frac{\text{A}}{\text{y}+1}+\frac{\text{B}}{\text{y}+2}$
$\Rightarrow1=\text{A}(\text{y}+2)+\text{B}(\text{y}+1)=(\text{A}+\text{B})\text{y}+(2\text{A}+\text{B})$
Equating similar terms, we get,
A + B = 0, and 2A + B = 1
Solving, we get,
Thus,
$\text{I}=\int\frac{\text{dx}}{\text{x}^2+1}-\int\frac{\text{dx}}{\text{x}^2+2}$
$\text{I}=\tan^{-1}\text{x}-\frac{1}{\sqrt{2}}\tan^{-1}\frac{\text{x}}{\sqrt{2}}+\text{C}$

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