Question

Integrate the following functions.
$\int x\sqrt{x+2}dx.$

Answer 1

$\int x\sqrt{x+2}dx=\int (x+2-2)\sqrt{x+2}dx=\int (x+2)\sqrt{x+2}dx-2\int \sqrt{x+2}dx=\int (x+2{)}^{\frac{3}{2}}dx-2\int (x+2{)}^{\frac{1}{2}}dx=\frac{(x+2{)}^{(\frac{3}{2}+1)}}{(\frac{3}{2}+1)}-2\frac{(x+2{)}^{(\frac{1}{2}+1)}}{\left(\frac{1}{2}\right)+1}+C[\because \int x^{n}dx=\frac{x^{n+1}}{n+1}]=\frac{2}{5}(x+2{)}^{\frac{5}{2}}-\frac{2\times 2}{3}(x+2{)}^{\frac{3}{2}}+C=\frac{2}{5}(x+2{)}^{\frac{5}{2}}(x+2{)}^{\frac{5}{2}}-\frac{4}{3}(x+2{)}^{\frac{3}{2}}+C$

Alternate MethodLet x+2 =t or $x=(t-2)\Rightarrow 1=\frac{dt}{dx}\Rightarrow dx=dt$
Then, $\int x\sqrt{x+2}dx=\int (t-2)\sqrt{t}dt=\int (t^{\frac{3}{2}}-2t^{\frac{1}{2}})dt$
$\frac{2}{5}(x+2{)}^{\frac{5}{2}}-\frac{4}{3}(x+2{)}^{\frac{3}{2}}+C$