Question

Evaluate ${\int }_0^{1}x{(1-x)}^{9}dx$

Answer 1

Apply integration by parts.

Let $I={\int }_0^{1}x{(1-x)}^{9}dx$

$={\left|\frac{x{(1-x)}^{10}}{10}\right|}_0^1-{\int }_0^1\frac{{(1-x)}^{10}}{10}dx$

$=0-{\left|\frac{(1-x{)}^{11}}{10\times 11}\right|}_0^1$

$=\frac{1}{110}$

Hence, $I={\int }_0^{1}x{(1-x)}^{9}dx=\frac{1}{110}$