The value of \(\int_{0}^{1} x^{2}\left(1-x^{2}\right)^{3 / 2} d x\) is

Let \(I=\int_{0}^{1} x^{2}\left(1-x^{2}\right)^{3 / 2} d x\)
\(\text {(let } x=\sin \theta \Rightarrow d x=\cos \theta d \theta) \)
\( I=\int_{0}^{\pi / 2} \sin ^{2} \theta\left(\cos ^{2} \theta\right)^{3 / 2} \cdot \cos \theta d \theta \)
\(=\int_{0}^{\pi / 2} \sin ^{2} \theta \cdot \cos ^{4} \theta d \theta\)
By Gamma function, \(I=\frac{\sqrt{\frac{2+1}{2}} \mid \frac{4+1}{2}}{2 \mid \frac{2+4+2}{2}}=\frac{\sqrt{\frac{3}{2}} \sqrt{\frac{5}{2}}}{2 \Gamma 4}\)
\(=\frac{\frac{1}{2} \mid \frac{1}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{\frac{1}{2}}}{2 \cdot 3 \cdot 2}\)
\(=\frac{3 \cdot \sqrt{\pi} \cdot \sqrt{\pi}}{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 2}=\frac{\pi}{32}\)