Question:
Evaluate: $\int(1-\mathrm{x}) \sqrt{\mathrm{x}} \mathrm{dx}$.
Solution:
Given, $\int(1-x) \sqrt{x} \mathrm{dx}$
$=\int(\sqrt{x}-x \sqrt{x}) d x$
$=\int\left(x^{\frac{1}{2}}-x \cdot x^{\frac{1}{2}}\right) d x$
$=\int x^{\frac{1}{2}}-x^{\frac{3}{2}} d x$
$=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+c\left[\right.$ since, $\left.\int x^{n} d x=\frac{x^{n+1}}{n+1}\right]$
$=\frac{x^{\frac{3}{2}}}{\frac{3}{2}}-\frac{x^{\frac{5}{2}}}{\frac{5}{2}}+c$
$=\frac{2}{3} x^{\frac{3}{2}}-\frac{2}{5} x^{\frac{5}{2}}+c$