Question

Solve:
$\int (x+2)\sqrt{3x+5}dx$

Answer 1

$I=\int (x+2)\sqrt{3x+5}dx$

Let $u=3x+5\Rightarrow dx=\frac{1du}{3}$

$I=\int (\frac{u-5}{3}+2)\sqrt{u}\left(du/3\right)$

$=\frac{1}{9}\int (u^{3/2}+u^{1/2})du$

$=\frac{1}{9}(\frac{2}{5}{u}^{5/2}+\frac{2}{3}{u}^{3/2})+c$

$I=\frac{1}{9}\left[\right(3x+5{)}^{5/2}\times \frac{2}{5}+\frac{2}{3}(3x+5{)}^{3/2}]+c$

$I=\frac{2(3x+5{)}^{3/2}(9x+20)}{135}+c$