Evaluate : `int_0^1x(1-x)^5dx`

Options

A. `(1)/(5)`
B. `(1)/(42)`
C. `(1)/(13)`
D. `(13)/(42)`

Answer

B

Explanation

Let `l=int_(0)^(1)x(1-x)^(5)dx`
put `1-x=timpliesx=1-t`
`implies-dx=d t impliesdx=-dt`
when x=0, then t=1 and
when x=1, then t=0
`thereforel=int_(1)^(0)(1-t)(t)^(5)(-d t)=int_(0)^(1)(t^(5)-t^(6))d t`
`=((t^(6))/(6)-(t^(7))/(7))_(0)^(1)=(1)/(6)(1)^(6)-(1)/(7)(1)^(7)`
`=(7-6)/(42)=(1)/(42)`