\(\int_{0}^{a}(a-x)^{\frac{3}{2}} \cdot x^{2} d x=\)

Let
\(I =\int_{0}^{a}(a-x)^{\frac{3}{2}} x^{2} d x \)
\( =\int_{0}^{a}[a-(a-x)]^{\frac{3}{2}}(a-x)^{2} d x \)
\( =\int_{0}^{a} x^{\frac{3}{2}}\left(a^{2}-2 a x+x^{2}\right) d x=\int_{0}^{a}(a^{2} x^{\frac{3}{2}}-\) \(2 a x^{\frac{5}{2}}+x^{\frac{7}{2}})_{d x} \)
\( =a^{2}\left[\frac{x^{\frac{5}{2}}}{\frac{5}{2}}\right]_{0}^{a}-2 a\left[\frac{x^{\frac{7}{2}}}{\frac{7}{2}}\right]_{0}^{a}+\left[\frac{x^{\frac{9}{2}}}{\frac{9}{2}}\right]_{0}^{a} \)
\( =\frac{2 a^{2}}{5}\left[a^{\frac{5}{2}}-0\right]-2 a \times \frac{2}{7}\left[a^{\frac{7}{2}}-0\right]+\frac{2}{9}[a^{\frac{9}{2}}-\) \(0]=\frac{2}{5} a^{\frac{9}{2}}-\frac{4}{7} a^{\frac{9}{2}}+\frac{2}{9} a^{\frac{9}{2}} \)
\( =a^{\frac{9}{2}}\left[\frac{2}{5}-\frac{4}{7}+\frac{2}{9}\right]=\frac{16}{315} a^{\frac{9}{2}} \)