Question

Solve : ${\int }_0^{2}x\sqrt{x+2}dx$

Answer 1

${\int }_0^{2}x\sqrt{x+2}dx$
Integration by parts
$=x\int \sqrt{x+2}dx-\int \frac{dx}{dx}(\int \sqrt{x+2}dx)dx$
$=x\frac{{(x+2)}^{3/2}}{3/2}-\int \frac{{(x+2)}^{3/2}}{3/2}dx$
$=2x\frac{{(x+2)}^{3/2}}{3}-\frac{2}{3}\frac{{(x+2)}^{5/2}}{5/2}{]}_0^2$
$=2x\frac{{(x+2)}^{3/2}}{3}-\frac{4{(x+2)}^{5/2}}{15}{]}_0^2$
$=\frac{4{\left(4\right)}^{3/2}}{3}-\frac{4{\left(4\right)}^{5/2}}{15}-0+\frac{4{\left(2\right)}^{5/2}}{15}$
$=\frac{32}{3}-\frac{4\times 32}{15}+\frac{16\sqrt{2}}{15}$
$=\frac{160}{15}-\frac{128}{15}+\frac{16\sqrt{2}}{15}$
$=\frac{32}{15}+\frac{16\sqrt{2}}{15}=\frac{16}{15}(2+\sqrt{2})$