Question

${\int }_0^{1}x{(1-x)}^{4}dx=\frac{1}{C}$, then $C=?$

Answer 1

Let $I={\int }_0^{1}x{(1-x)}^{4}dx$
Using ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$
$\therefore I={\int }_0^1(1-x)x^{4}dx={\int }_0^1(x^4-x^5)dx=\frac{1}{5}-\frac{1}{6}=\frac{1}{30}$