`int_(a^4)^(b^4)(f(sqrtx))/((sqrtx)[f(a^2+b^2-sqrtx)+f(sqrtx)])dx` is equal to

Options

A. `a^2-b^2`
B. `b^2-a^2`
C. 0
D. `a^2+b^2`

Answer

B

Explanation

Let `sqrt(x)=t` also `(1)/(2sqrt(x))dx=dt implies x=t^(2)`
When `x=a^(4) implies t=a^(2)`
`x=b^(4) implies t=b^(2)`
`implies I=2int_(a^(2))^(b^(2))(f(t))/((f(a^(2)+b^(2)-t)+f(t)))dt` . . (1)
`implies I=2int_(a^(2))^(b^(2)) (f(a^(2)+b^(2)-t))/(f(t)+f(a^(2)+b^(2)-t)) dt ` . . . (2)
Adding 2I=2`int_(a^(2))^(b^(2))dt" "I=(b^(2)-a^(2))`.