`int_(0)^(a) x (1-x)^(5)dx=........`

Options

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

Answer

B

Explanation

Let `I = int_(0)^(1) x (1 - x)^(5) dx`
put `1 - x = t implies x = 1 - t`
`implies - dx = dt`
`implies dx = - dt`
When x = 0, then t = 1 and
When x = 1, then t = 0
`:. I = int_(1)^(0) (1 - t) (t)^(5) (-dt) = int_(0)^(1) (t^(5) - t^(6)) dt`
`= ((t^(6))/(6) - (t^(7))/(7))_(0)^(1) = (1)/(6) (1)^(6) - (1)/(7) (1)^(7)`
`= (7 - 6)/(42) = (1)/(42)`