Question

Solve : $\underset{0}{\stackrel{1}{\int }}{x}^2(1-x^2{)}^{3/2}dx=$

Answer 1

${\int }_0^1{x}^2(1-x^2{)}^{\frac{3}{2}}dx$

$u=(1-x^2{)}^{\frac{3}{2}},v^{\prime }=x^2$

$\int x^2(1-x^2{)}^{\frac{3}{2}}dx=\frac{1}{3}{x}^3(1-x^2{)}^{\frac{3}{2}}-\int -x^4\sqrt{1-x^2}dx$

$=\frac{1}{3}{x}^3(1-x^2{)}^{\frac{3}{2}}-\left(\frac{1}{6}{x}^3\right(\sqrt{1-x^2}{)}^3-\frac{1}{16}{\sin }^{-1}x+\frac{1}{64}\sin \left(4{\sin }^{-1}x\right))$

$=\frac{1}{3}{x}^3(1-x^2{)}^{\frac{3}{2}}-\frac{1}{6}{x}^3(\sqrt{1-x^2}{)}^3+\frac{1}{16}{\sin }^{-1}x-\frac{1}{64}\sin \left(4{\sin }^{-1}x\right)+C$

${\int }_0^1{x}^2(1-x^2{)}^{\frac{3}{2}}dx=\frac{\pi }{32}-0$

${\int }_0^1{x}^2(1-x^2{)}^{\frac{3}{2}}dx=\frac{\pi }{32}$