Question

Integrate : $\underset{0}{\overset{1}{\int }}x{\left(1-x\right)}^{91}$

Answer 1

Let $t=1-x\Rightarrow dt=-dx$

When $x=0\Rightarrow t=1$

When $x=1\Rightarrow t=0$

$=-{\int }_1^0{t}^{91}(1-t)dt$

$=-{\int }_1^0{t}^{91}dt+{\int }_1^0{t}^{92}dt$

$={[-\frac{t^{91+1}}{92}+\frac{t^{92+1}}{93}]}_1^0$

$={[-\frac{t^{92}}{92}+\frac{t^{93}}{93}]}_1^0$

$=-\frac{1}{92}(0-1)+\frac{1}{93}(0-1)$

$=\frac{1}{92}-\frac{1}{93}=\frac{93-92}{92\times 93}=\frac{1}{8,556‬}$