Evaluate the following integrals: 2x^4+7x^3+6x^2 x^2+2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{2\text{x}^4+7\text{x}^3+6\text{x}^2}{\text{x}^2+2\text{x}}\text{dx}$

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Answer

$\int\bigg(\frac{2\text{x}^4+7\text{x}^3+6\text{x}^2}{\text{x}^2+2\text{x}}\bigg)\text{dx}$
$=\int\frac{\text{x}^2(2\text{x}^2+7\text{x}+6}{\text{x}(\text{x}+2)}$
$=\int\frac{\text{x}\big[2\text{x}^2+4\text{x}+3\text{x}+6\big]}{(\text{x}+2)}\text{dx}$
$=\int\frac{\text{x}(2\text{x}(\text{x}+2)+3(\text{x}+2))}{(\text{x}+2)}\text{dx}$
$=\int\frac{\text{x}(2\text{x}+3)(\text{x}+2)}{(\text{x}+2)}\text{dx}$
$=\int(2\text{x}^2+3\text{x})\text{dx}$
$=2\int\text{x}^2\text{dx}+3\int\text{x }\text{dx}$
$=2\Big[\frac{\text{x}^3}{3}\Big]+3\Big[\frac{\text{x}^2}{2}\Big]+\text{C}$
$=\frac{2}{3}\text{x}^3+\frac{3}{2}\text{x}^2+\text{C}$

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