Question
Evaluate the following intregals:
$\int\frac{2\text{x}^2+7\text{x}-3}{\text{x}^2(2\text{x}+1)}\text{ dx}$
$\int\frac{2\text{x}^2+7\text{x}-3}{\text{x}^2(2\text{x}+1)}\text{ dx}$
$\therefore\frac{2\text{x}^2+7\text{x}-3}{\text{x}^2(2\text{x}+1)}=\frac{13}{\text{x}}-\frac{3}{\text{x}^2}-\frac{24}{2\text{x}+1}$
$\Rightarrow\text{I}=13\int\frac{\text{dx}}{\text{x}}-3\int\text{x}^{-2}\text{dx}-24\int\frac{\text{dx}}{2\text{x}+1}$
$=13\log|\text{x}|+\frac{3}{\text{x}}-24\frac{\log|2\text{x}+1|}{2}+\text{C}$
$=13\log|\text{x}|+\frac{3}{\text{x}}-12\log|2\text{x}-1|+\text{C}$
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$(\text{xy}^2+2\text{x})\text{dx}+(\text{x}^2\text{y+2y})\text{dy}=0$
$\text{x}+\text{y}\geq8$
$\text{x}+4\text{y}\geq12$
$\text{x}\geq3,\text{y}\geq2$