Question

Find the integrals (primitives):
$\rm \int x\sqrt{x+2}\ dx$

Answer 1

here is the answer ☺️❤️

Answer 2

For integration we first simplify the expression

$let \: x + 2 = u \\ \\ dx = du \\ \\ x = u - 2$
$\int x\sqrt{x+2}\ dx \\ \\ \int (u - 2) \sqrt{u} \: du \\ \\ \\ \int (u \sqrt{u} - 2\sqrt{u} )\: du \\ \\ \\ \int {u}^{ \frac{3}{2}}du -2\int \sqrt{u} \: du \\ \\ = \frac{ {u}^{ \frac{5}{2} } }{ \frac{5}{2} } - 2 \frac{ {u}^{ \frac{3}{2} } }{ \frac{3}{2} }+C \\ \\ = \frac{2}{5} {u}^{ \frac{5}{2}} - \frac{4}{3} {u}^{ \frac{4}{3}} + C \\ \\ redo \: substitution \\ \\ \int x\sqrt{x+2}\ dx = \frac{2}{5} {(x + 2)}^{ \frac{5}{2}} - \frac{4}{3} {(x + 2)}^{ \frac{4}{3}} + C$
Hope it helps you.