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

Question

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

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Answer

Let $\frac{2\text{x}+1}{(\text{x}-2)(\text{x}-3)}=\frac{\text{A}}{(\text{x}-2)}+\frac{\text{B}}{(\text{x}-3)}$
$\Rightarrow2\text{x}+1=\text{A}(\text{x}-3)+\text{B}(\text{x}-2)$
$=(\text{A}+\text{B})\text{x}+(-3\text{A}-2\text{B})$
equating similar terms, we get
A + B = 2, and -3A - 2B = 1
Thus,
$\text{I}=-5\int\frac{\text{dx}}{\text{x}-2}+7\int\frac{\text{dx}}{\text{x}-3}$
$=-5\log|\text{x}-2|+7\log|\text{x}-3|+\text{C}$
$\text{I}=\log\Big|\frac{(\text{x}-3)^7}{(\text{x}-2)^5}\Big|+\text{C}$

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