Question

$\int 4x^3\sqrt{5-x^2}dx$

Answer 1

$$\begin{gathered} \int 4x^3\sqrt{5-x^2}dx \\ =4\int x^2\times x\sqrt{5-x^2}dx \\ \mathrm{Let}5-x^2=t \\ \Rightarrow x^2=5-t \\ \Rightarrow 2x=-\frac{dt}{dx} \\ \Rightarrow xdx=-\frac{dt}{2} \\ \mathrm{Now},4\int x^2\times x\sqrt{5-x^2}dx \\ =\frac{4}{-2}\int \left(5-t\right).\sqrt{t}dt \\ =-2\int 5t^{\frac{1}{2}}+2\int t^{\frac{3}{2}}dt \\ =-10\left[\frac{t^{{\displaystyle \frac{1}{2}}+1}}{{\displaystyle \frac{1}{2}}+1}\right]+2\left[\frac{t^{{\displaystyle \frac{3}{2}}+1}}{{\displaystyle \frac{3}{2}}+1}\right]+C \\ =-\frac{20}{3}{t}^{\frac{3}{2}}+\frac{4}{5}{t}^{\frac{5}{2}}+C \\ =-\frac{20}{3}{\left(5-x^2\right)}^{\frac{3}{2}}+\frac{4}{5}{\left(5-x^2\right)}^{\frac{5}{2}}+C \end{gathered}$$