x^2+x+5 3x+2 dx - Vidyadip

Question

$\int\frac{\text{x}^2+\text{x}+5}{3\text{x}+2}\text{dx}$

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Answer

$\int\frac{(\text{x}^2+\text{x}+5)}{(3\text{x}+2)}\text{dx}$
$=\frac{1}{9}\int\frac{9\text{x}^2+9\text{x}+45}{(3\text{x}+2)}\text{dx}$
$=\frac{1}{9}\Big[\int\frac{9\text{x}^2-4}{3\text{x}+2}\text{dx}+\int\frac{9\text{x}+6}{3\text{x}+2}\text{dx}+\int\frac{43}{3\text{x}+2}\text{dx}\Big]$
$=\frac{1}{9}\Big[\int\frac{(3\text{x}-2)(3\text{x}+2)}{(3\text{x}+2)}\text{dx}+\int\frac{3(3\text{x}+2)}{3\text{x}+2}\text{dx}+43\int\frac{\text{dx}}{3\text{x}+2}\Big]$
$=\frac{1}{9}\Big[\int(3\text{x}-2)\text{dx}+3\int1\text{dx}+43\int\frac{\text{dx}}{3\text{x}+2}\Big]$
$=\frac{1}{9}\Big[\Big(3\frac{\text{x}^2}{2}-2\text{x}\Big)+3\text{x}+\frac{43}{3}\text{ln}|3\text{x}+2|+\text{c}\Big]$
$=\frac{1}{9}\Big[\frac{3}{2}\text{x}^2+\text{x}-\frac{43}{3}\text{ln}|3\text{x}+2|+\text{c}\Big]$
$=\frac{1}{6}\text{x}^2+\frac{1}{9}\text{x}-\frac{43}{27}\text{ln}|3\text{x}+2|+\text{c}$

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