Explanation:
(A) : Given,\(\int_{0}^{3} x(1-x)^{3 / 2} d x\)
Let, \(1-\mathrm{x}=\mathrm{t}\)
\(\mathrm{x}=(1-\mathrm{t})\)
\(\mathrm{dx}=-\mathrm{dt}\)
If, \(x=0, t=1\)
If, \(x=1, t=0\)
Let, \(\quad \mathrm{I}=-\int_{1}^{0}(1-\mathrm{t}) \times(\mathrm{t})^{3 / 2} \mathrm{dt}\)
\(I=-\int_{1}^{0}\left\{(t)^{3 / 2}-(t)^{5 / 2}\right\} d t\)
\(=-\left[\frac{2}{5}(\mathrm{t})^{5 / 2}-\frac{2}{7}(\mathrm{t})^{7 / 2}\right]_{1}^{0}\)
\(=-\left[\left(\frac{2}{5} \times 0-\frac{2}{7} \times 0\right)-\left(\frac{2}{5} \times 1-\frac{2}{7} \times 1\right)\right]\)
\(=\left[-\frac{2}{5}+\frac{2}{7}\right]=-\left[\frac{-14+10}{35}\right]=\frac{4}{35}\)