Integrate the function: x x+2 - Vidyadip

Question

Integrate the function: $x \sqrt{x+2}$

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Answer

Let I = $ \int x \sqrt{x+2} d x$
Put (x + 2) = t $\Rightarrow$ dx = dt
$\Rightarrow I = \int x \sqrt{x+2} d x=\int(t-2) \sqrt{t} d t$
$=\int\left(t^{\frac{3}{2}}-2 t^{\frac{t}{2}}\right) d t$
$=\int t^{\frac{3}{2}} d t-2 \int t^{\frac{1}{2}} d t$
$=\frac{t^{\frac{5}{2}}}{\frac{5}{2}}-2\left(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right)+C$
$=\frac{2}{5} t^{\frac{5}{2}}-\frac{4}{3} t^{\frac{3}{2}}+C$
$=\frac{2}{5}(x+2)^{\frac{5}{2}}-\frac{4}{3}(x+2)^{\frac{3}{2}}+C$

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