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

Question

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

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Answer

$\int\frac{(\text{x}^2+5\text{x}+2)}{(\text{x}+2)}\text{dx}$
$=\int\frac{\text{x}^2}{\text{x}+2}\text{dx}+5\int\frac{\text{x dx}}{\text{x}+2}+2\int\frac{\text{dx}}{\text{x}+2}$
$=\int\Big(\frac{\text{x}^2-4+4}{\text{x}+2}\Big)\text{dx}+5\int\Big(\frac{\text{x}+2-2}{\text{x}+2}\Big)\text{dx}+2\int\frac{\text{dx}}{\text{x}+2}$
$=\int\frac{(\text{x}-2)(\text{x}+2)}{(\text{x}+2)}\text{dx}+\int\frac{4}{\text{x}+2}\text{dx}+5\int\Big(1-\frac{2}{\text{x}+2}\Big)\text{dx}+2\int\frac{\text{dx}}{\text{x}+2}$
$=\int(\text{x}-2)\text{dx}+4\int\frac{\text{dx}}{\text{x}+2}+5\int\text{dx}-10\int\frac{\text{dx}}{\text{x}+2}+2\int\frac{\text{dx}}{\text{x}+2}$
$=\int(\text{x}-2)\text{dx}-4\int\frac{\text{dx}}{\text{x}+2}+5\int\text{dx}$
$=\Big(\frac{\text{x}^2}{2}-2\text{x}\Big)-4\text{ln|}\text{x}+2|+5\text{x}+\text{C}$
$=\frac{\text{x}^2}{2}+3\text{x}-4\text{ln}|\text{x}+2|+\text{C}$

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