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

Question

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

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Answer

Let $\int\frac{2\text{x}+1}{(\text{x}+1)(\text{x}-2)}=\frac{\text{A}}{(\text{x}+1)}+\frac{\text{B}}{(\text{x}-2)}$$\Rightarrow2\text{x}+1=\text{A}(\text{x}-2)+\text{B}(\text{x}+1)$
Put $\text{x}=2$
$\Rightarrow5=3\text{B}\Rightarrow\text{B}=\frac{5}{3}$
Put $\text{x}=-1$
$\Rightarrow-1=-3\text{A}\Rightarrow\text{A}=\frac{1}{3}$
so,
$\int\frac{2\text{x}+1}{(\text{x}+1)(\text{x}-2)}\text{ dx}=\frac{1}{3}\int\frac{\text{dx}}{\text{x}+1}+\frac{5}{3}\int\frac{\text{dx}}{\text{x}-2}$
$=\frac{1}{3}\log|\text{x}+1|+\frac{5}{3}\log|\text{x}-2|+\text{C}$
Thus
$\text{I}=\frac{1}{3}\log|\text{x}+1|+\frac{5}{3}\log|\text{x}-2|+\text{C}$

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