Question

$\int x\sqrt{x+2}dx$

Answer 1

$\mathrm{Let}I=\int x\sqrt{x+2}dx$
Putting x + 2 = t
Then, x = t – 2
Difference both sides
dx = dt
Now, integral becomes
$$\begin{gathered} I=\int \left(t-2\right)\sqrt{t}dt \\ =\int \left(t^{\frac{3}{2}}-2t^{\frac{1}{2}}\right)dt \\ =\left[\frac{t^{{\displaystyle \frac{3}{2}+1}}}{{\displaystyle \frac{3}{2}+1}}-2\frac{t^{{\displaystyle \frac{1}{2}+1}}}{{\displaystyle \frac{1}{2}+1}}\right]+\mathrm{C} \\ =\frac{2}{5}{t}^{\frac{5}{2}}-\frac{4}{3}{t}^{\frac{3}{2}}+\mathrm{C} \\ =\frac{2}{5}{\left(x+2\right)}^{\frac{5}{2}}-\frac{4}{3}{\left(x+2\right)}^{\frac{2}{3}}+\mathrm{C} \end{gathered}$$