x+2 dx - Vidyadip

Question

$\int\text{x}\sqrt{\text{x}+2}\ \text{dx}$

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Answer

Let I $=\int\text{x}\sqrt{\text{x}+2}\text{ dx}.$ Then,
$\text{I}=\int\{(\text{x}+2)-2\}\text{x}+2\text{dx}\ \ \ [\because\text{x}=(\text{x}+2)-2]$
$\Rightarrow\text{I}=\int\Big\{(\text{x}+2)^\frac{3}{2}-2(\text{x}+2)^\frac{1}{2}\Big\}\text{dx}$
$\Rightarrow\text{I}=\frac{2}{5}(\text{x}+2)^\frac{5}{2}-\frac{4}{3}(\text{x}+2)^\frac{3}{2}+\text{C}$

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