Given a real-valued function f which is monotonic and differentiable. Then `int_(f(a))^(f(b))2x(b-f^(-1)(x))dx=`

Given a real valued fuction f(x) which is monotonic and differentiable , prove that for any real number a and b, int_(a)^(b){f^(2)(x)-f^(2)(a)} dx = int_(f(a))^(f(b))2x{b-f^(-1)(x)} dx

If f(x) is monotonic differentiable function on [a , b] , then int_a^bf(x)dx+int_(f(a))^(f(b))f^(-1)(x)dx= (a) bf(a)-af(b) (b) bf(b)-af(a) (c) f(a)+f(b) (d) cannot be found

f is a real valued function from R to R such that f(x)+f(-x)=2 , then int_(1-x)^(1+X)f^(-1)(t)dt=

If int_(a)^(b)(f(x))/(f(x)+f(a+b-x))dx=10 , then

Prove that int_(a)^(b)f(x)dx=(b-a)int_(0)^(1)f((b-a)x+a)dx

Prove that: int_a^b(f(x))/(f(x)+f(a+b-x))dx=(b-a)/2

Let f: R → R be a one-one onto differentiable function, such that f(2)=1 and f^(prime)(2)=3. Then, find the value of (d/(dx)(f^(-1)(x)))_(x=1)

Let f: [a, b] -> R be a function, continuous on [a, b] and twice differentiable on (a, b) . If, f(a) = f(b) and f'(a) = f'(b) , then consider the equation f''(x) - lambda (f'(x))^2 = 0 . For any real lambda the equation hasatleast M roots where 3M + 1 is

Let f(x) be a differentiable function in the interval (0, 2) then the value of int_(0)^(2)f(x)dx

If f is an odd function, then evaluate I=int_(-a)^a(f(sinx)dx)/(f(cosx)+f(sin^2x))