Question

Solve:

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

Answer 1

Solution

$I={\int }_0^{1}x.(1-x{)}^{5}dx$

${\int }_a^{b}f\left(x\right)={\int }_a^{b}f(a+b-x)dx$

$\therefore I={\int }_0^1(1-x)(1-(1-x){)}^{5}dx$

$={\int }_0^1(1-x)x^{5}dx$

$={\int }_0^1(x^5-x^6)dx$

$={\left[\frac{x^6}{6}-\frac{x^7}{7}\right]}_0^1=\frac{1}{6}-\frac{1}{7}=\frac{1}{42}$