What is `int_(0)^(1)x (1-x)^(9) dx` equal to ?

Options

A. `1/110`
B. `1/111`
C. `1/112`
D. `1/119`

Answer

A

Explanation

Let the given integral be,
`I = int_(0)^(1) x(1-x)^(9) dx`
Put `1 -x = t rArr dx = - dt `and `x = 1- t`
when ` = 0. t = 1` and when `x = 1, t = 0`
`rArr I = int_(1)^(0) (1-t)t^(9)(-dt)`
`= - int_(1)^(0)(-t^(10) + t^(9)) dt = - int_(1)^(0) (-t^(10) + t^(9)) dt`
`[(-t^(11))/(11) +(t^(10))/(10)]_(0)^(1) = (-1)/(11) + (1)/(10) =(-10 + 11)/(110) = (1)/(110)`