x^3 x-2 dx - Vidyadip

Question

$\int\frac{\text{x}^3}{\text{x}-2}\text{dx}$

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Answer

$\text{Let I}=\int\frac{\text{x}^3}{\text{x}-2}\text{dx}$
Using long division method, we have
$\frac{\text{x}^3}{\text{x}-2}=\text{x}^2+2\text{x}+4+\frac{8}{\text{x}-2}$
$\text{I}=\int\Big(\text{x}^2+2\text{x}+4+\frac{8}{\text{x}-2}\Big)\text{dx}$
$=\int\text{x}^2\text{dx}+2\int\text{xdx}+4\int\text{dx}+8\int\frac{1}{\text{x}-2}\text{dx}$
$=\frac{\text{x}^3}{3}+\frac{2\text{x}^2}{2}+4\text{x}+8\log|\text{x}-2|+\text{c}$
$=\frac{\text{x}^3}{3}+\text{x}^2+4\text{x}+8\log|\text{x}-2|+\text{c}$
$\therefore\text{I}=\frac{\text{x}^3}{3}+\text{x}^2+4\text{x}+8\log|\text{x}-2|+\text{c}$

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