Question

The value of the integral `overset(1)underset(0)int x(1-x)^(n)dx`, is
A. `1/(n+1)`
B. `1/(n+2)`
C. `1/(n+2)-1/(n+2)`
D. `1/(n+1) +1/(n+2)`

Answer 1

Correct Answer - C
Let `l=int_(0)^(1)xx(1-x)^(n)dx`
`therefore -i=int_(0)^(1)-x (1-x)^(n)dx=int_(0)^(1)(1-x-1)(1-x)^(n)dx`
`=int_(0)^(1)(1-x)^(n+1)dx-int_(0)^(1)(1-x)^(n)dx`
`=[((1-x)^(n+2))/(-(n+))]_(0)^(1)-[((1-x)^(n+1))/(-(n+1))]_(0)^(1)`
`=(1)/(n+2)-(1)/(n+1)`
`therefore l=(1)/(n+1)-(1)/(n+2)`.