Let `f` be a continuous function on `[a ,b]dot` Prove that there exists a number `x in [a , b]` such that `int_a^xf(t)dx=int_x^bf(t)dtdot`

Prove that int_-a^a xf(x^4)dx=0

Let f be continuous on [a , b],a >0,a n d differentiable on (a , b)dot Prove that there exists c in (a , b) such that (bf(a)-af(b))/(b-a)=f(c)-cf^(prime)(c)

Let f be continuous on [a , b],a >0,a n d differentiable on (a , b)dot Prove that there exists c in (a , b) such that (bf(a)-af(b))/(b-a)=f(c)-cf^(prime)(c)

Let f be a continuous function on [a , b]dot If F(x)=(int_a^xf(t)dt-int_x^bf(t)dt)(2x-(a+b)), then prove that there exist some c in (a , b) such that int_a^cf(t)dt-int_c^bf(t)dt=f(c)(a+b-2c)dot

STATEMENT 1: If f(x) is continuous on [a , b] , then there exists a point c in (a , b) such that int_a^bf(x)dx=f(c)(b-a) STATEMENT 2: For a < b , if ma n dM are, respectively, the smallest and greatest values of f(x)on[a , b] , then m(b-a)lt=int_a^bf(x)dxlt=(b-a)Mdot

Let f(x) be a continuous function AAx in R , except at x=0, such that int_0^a f(x)dx , ain R^+ exists. If g(x)=int_x^a(f(t))/t dt , prove that int_0^af(x)dx=int_0^ag(x)dx

Let f(x) be a continuous function and I = int_(1)^(9) sqrt(x)f(x) dx , then

Let f(x) be a continuous function AAx in R , except at x=0, such that int_x^a(f(t))/t dt ,p rov et h a tint_0^af(x)dx=int_0^ag(x)dx

Let f(x) be a continuous function in R such that f(x) does not vanish for all x in R . If int_1^5 f(x)dx=int_-1^5 f(x)dx , then in R, f(x) is (A) an even function (B) an odd function (C) a periodic function with period 5 (D) none of these

If f is an integrable function, show that int_(-a)^a xf(x^2)dx=0