Question

Integrate the functions:

${\displaystyle x\sqrt{x+2}}$

Answer 1

Let ${\displaystyle I=\int x\sqrt{x+2}}$ dx

Taking ${\displaystyle \sqrt{x+2}=t}$

or x + 2 = t²

or dx = 2t dt

Hence, ${\displaystyle I=\int (t^2-2).t.2tdt}$

${\displaystyle =\int 2t^4-4t^{2}dt}$

${\displaystyle =2\int (t^4-2t^2)}$ dt

${\displaystyle =2\int t^{4}dt-4\int }$ t² dt

${\displaystyle =\frac{2}{5} t^5-\frac{4}{3} t^3+C}$

${\displaystyle \Rightarrow \frac{2}{5}{(x+2)}^{\frac{5}{2}}-\frac{4}{3}{(x+2)}^{\frac{3}{2}}+C}$