Evaluate the following intregals: x^2+1 (x-2)^2(x+3) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{\text{x}^2+1}{(\text{x}-2)^2(\text{x}+3)}\ \text{dx}$

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Answer

Let $\int\frac{\text{x}^2+1}{(\text{x}-2)^2(\text{x}+3)}=\frac{\text{A}}{\text{x}-2}+\frac{\text{B}}{(\text{x}-2)^2}+\frac{\text{C}}{\text{x}+3}$

$\Rightarrow\text{x}^2+1=\text{A}(\text{x}-2)(\text{x}+3)(\text{x}+3)+\text{B}(\text{x}+3)+\text{C}(\text{x}-2)^2$

$=(\text{A}+\text{C})\text{x}^2+(\text{A}+\text{B}-4\text{C})\text{x}+(-6\text{A}+3\text{B}+4\text{C})$

Equating similar terms, we get,

A + C = 1, A + B - 4C = 0, -6A + 3B + 4C = 1

Solving we get, $\text{A}=\frac{3}{5},\text{B}=1,\text{C}=\frac{2}{5}$

Thus,

$\text{I}=\frac{3}{5}\int\frac{\text{dx}}{\text{x}-2}+\int\frac{\text{dx}}{(\text{x}-2)^2}+\frac{2}{5}\int\frac{\text{dx}}{\text{x}+3}$

$\text{I}=\frac{3}{5}\log\text{x}-2|-\frac{1}{(\text{x}-2)}+\frac{2}{5}\log|\text{x}+3|+\text{C}$

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