The value of `int_(1//n)^((an-1)//n) (sqrt(x))/(sqrt(a-x)+sqrtx)dx`, is

Options

A. `(a)/(2)`
B. `(1)/(2n)(na+2)`
C. `(na-2)/(2n)`
D. none of these

Answer

C

Explanation

Let
`underset(1//n)overset((an-1)//n)int(sqrt(x))/(sqrt(a-x+sqrtx))dx" "....(i)`
Using `underset(a)overset(b)intf(x)dx=underset(a)overset(b)intf(a+b-x)dx`, we have
`I=underset(1//n)overset((an-1)//n)int(sqrt(a-x))/(sqrt(x)+sqrt(a-x))dx" "....(ii)`
Adding (i) and (ii), we get
`2I=underset(1//n)overset((an-1)//n)int1.dx=(an-2)/(n)rArr I= (an-2)/(2n)`
ALTER Let `a=(1)/(n),b=(an-1)/(n)` and f'(x)`=sqrt(x)`. Then,
`I=underset(a)overset(b)int (f(x))/(f(a+b-x)+f(x))=(1)/(2)((an-1)/(n)-(1)/(n))=(an-2)/(2n)`