The value of `("^n C_0)/n`+`("^nC_1)/(n+1)`+`("^nC_2)/(n+2)`+....+`("^nC_ n)/(2n)` is equal to

Options

A. `underset(0)overset(1)intx^(n-1)(1-x)^(n)dx`
B. `underset(1)overset(2)intx^(n)(x-1)^(n-1)dx`
C. `underset(1)overset(2)int(1+x)^(n)dx`
D. `underset(0)overset(1)int(1-x)^(n)x^(n-1)dx`

Answer

B

Explanation

Let,
`S = (.^(n)C_(0))/(n)+(.^(n)C_(1))/(n+1)+(.^(n)C_(2))/(n+2)+"...."+(.^(n)C_(n))/(2n)`
`= .^(n)C_(0)underset(0)overset(1)intx^(n-1)+.^(n)C_(1)underset(0)overset(1)intx^(n)dx+"....."+.^(n)C_(n)underset(0)overset(1)intx^(2n-1)dx`
`=underset(0)overset(1)int[.^(n)C_(0)x^(n-1)+.^(n)C_(1)x^(n)+"..."+.^(n)C_(n)x^(2n-1)]dx`
`= underset(0)overset(1)intx^(n-1)(1+x)^(n)dx`
`= underset(1)overset(2)intx^(n)(x-1)^(n-1)dx`