`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 `sqrtx=t" also "(1)/(2sqrtx)dx=dt" "rArr" "x=t^(2)`
`"When "x=a^(4)" "rArr" "t=a^(2)," "x=b^(4)rArr t=b^(2)`
`rArr" "I=2 int_(a^(2))^(b^(2))(f(t))/((f(a^(2)+b^(2)-t)+f(t)))dt" (1)"`
`rArr" "I=2 int_(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))`