Evaluate the following intregals: 3+4x-x^2 (x+2)(x-1) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{3+4\text{x}-\text{x}^2}{(\text{x}+2)(\text{x}-1)}\ \text{dx}$

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Answer

$\text{I}=\int\frac{3+4\text{x}-\text{x}^2}{(\text{x}+2)(\text{x}-1)}\ \text{dx}$
$=\int-1+\frac{5\text{x}+1}{(\text{x}+2)(\text{x}-1)}\ \text{dx}$
$\Rightarrow\text{I}=-\int\text{dx}+\int\frac{5\text{x}+1}{(\text{x}+2)(\text{x}-1)}\ \text{dx}\ \dots(1)$
Let $\frac{5\text{x}+1}{(\text{x}+2)(\text{x}-1)}=\frac{\text{A}}{\text{x}+2}+\frac{\text{B}}{\text{x}-1}$
$\Rightarrow5\text{x}+1=\text{A}(\text{x}-1)+\text{B}(\text{x}+2)$
put x = 1
$\Rightarrow6=3\text{B}\Rightarrow\text{B}=2$
Put x = -2
$\Rightarrow-9=-3\text{A}\Rightarrow\text{A}=3$
So,
$\text{I}=\int\text{dx}+3\int\frac{\text{dx}}{\text{x}+2}+2\int\frac{\text{dx}}{\text{x}-1}$
$\text{I}=-\text{x}+3\log|\text{x}+2|+2\log|\text{x}-1|+\text{C}$

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