Question

$\underset{0}{\overset{1}{\int }}x{\left(1-x\right)}^{5}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I={\int }_0^{1}x{\left(1-x\right)}^5\mathit{d}x\mathit{.}\mathit{}\mathrm{Then}\mathit{,} \\ I={\int }_0^1\left(x-1+1\right){\left(1-x\right)}^5\mathit{d}x \\ \Rightarrow I={\int }_0^1\left[-{\left(1-x\right)}^6+{\left(1-x\right)}^5\right]\mathit{d}x \\ \Rightarrow I={\left[\frac{{\left(1-x\right)}^7}{7}\right]}_0^1-{\left[\frac{{\left(1-x\right)}^6}{6}\right]}_0^1 \\ \Rightarrow I=-\frac{1}{7}+\frac{1}{6} \\ \Rightarrow I=\frac{1}{42} \end{gathered}$$