Question

Evaluate: $\int \left(1-x\right)\sqrt{x}\mathrm{d}x$

Answer 1

$$\begin{gathered} \int \left(1-x\right)\sqrt{x}\mathrm{d}x=\int \left(\sqrt{x}-x\sqrt{x}\right)\mathrm{d}x \\ =\int \left(x^{\frac{1}{2}}-x^{\frac{3}{2}}\right)\mathrm{d}x \\ =\frac{x^{{\displaystyle \frac{1}{2}}+1}}{{\displaystyle \frac{1}{2}}+1}-\frac{x^{{\displaystyle \frac{3}{2}}+1}}{{\displaystyle \frac{3}{2}}+1}+c \\ =\frac{2}{{\displaystyle 3}}{x}^{\frac{3}{2}}-\frac{2}{{\displaystyle 5}}{x}^{\frac{5}{2}}+c \\ \mathrm{Hence},\int \left(1-x\right)\sqrt{x}\mathrm{d}x=\frac{2}{{\displaystyle 3}}{x}^{\frac{3}{2}}-\frac{2}{{\displaystyle 5}}{x}^{\frac{5}{2}}+c \end{gathered}$$