Question

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

Answer 1

$$\begin{gathered} \int x^2\sqrt{x+2}dx \\ \mathrm{Let}x+2=t \\ \Rightarrow x=t-2 \\ \Rightarrow dx=dt \\ \mathrm{Now},\int x^2\sqrt{x+2}dx \\ =\int {\left(t-2\right)}^2\sqrt{t}dt \\ =\int \left(4^2-4t+4\right)t^{\frac{1}{2}}dt \\ =\int \left(t^{2+\frac{1}{2}}-4t^{1+\frac{1}{2}}+4t^{\frac{1}{2}}\right)dt \\ =\int \left(t^{\frac{5}{2}}-4t^{\frac{3}{2}}+4t^{\frac{1}{2}}\right)dt \\ =\left[\frac{t^{{\displaystyle \frac{5}{2}}+1}}{{\displaystyle \frac{5}{2}}+1}\right]-4\left[\frac{t^{{\displaystyle \frac{3}{2}}+1}}{{\displaystyle \frac{3}{2}}+1}\right]+4\left[\frac{t^{{\displaystyle \frac{1}{2}}+1}}{{\displaystyle \frac{1}{2}}+1}\right]+C \\ =\frac{2}{7}{t}^{\frac{7}{2}}-\frac{8}{5}{t}^{\frac{5}{2}}+\frac{8}{3}{t}^{\frac{3}{2}}+C \\ =\frac{2}{7}{\left(x+2\right)}^{\frac{7}{2}}-\frac{8}{5}{\left(x+2\right)}^{\frac{5}{2}}+\frac{8}{3}{\left(x+2\right)}^{\frac{3}{2}}+C \end{gathered}$$