Question

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

Answer 1

Let (x + 2) = t

∴ dx = dt

$\Rightarrow \int x \sqrt{x+2}dx = \int (t-2) \sqrt{t}dt$

= $\int \bigg(t^{\frac{3}{2}}-2t^{\frac{1}{2}}\bigg)dt$

= $\int t^{\frac{3}{2}}dt-2\int t^{\frac{1}{2}}dt$

=$\frac{t^\frac{5}{2}}{\frac{5}{2}}-2\bigg(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\bigg)+C$

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

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