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`

Let f be a continuous function on [a,b]. Prove that there exists a number x in[a,b] such that int_(a)^(x)f(t)dx=int_(x)^(b)f(t)dt

Let f be a continous function on [a,b] . Prove that there exists a number x in [a, b] such that int _(a) ^(x) f(t) dt = int_(x) ^(b) f(t) dt .

If f is a continuous function on the interval [a,b] and there exists some c in(a,b) then prove that int_(a)^(b)f(x)dx=f(c)(b-a)

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

Let f be continuous on [a,b],a>0, and differentiable on (a,b). Prove that there exists c in(a,b) such that (bf(a,b))/(b-a)=f(c)-cf'(c)

Let f(x) be a continuous function AA x in R, except at x=0, such that int_(0)^(a)f(x)dxa in R^(+) exists.If g(x)=int_(x)^(a)(f(t))/(t)dt, prove that int_(0)^(a)f(x)dx=int_(0)^(a)g(x)dx

If f(t) is a continuous function defined on [a,b] such that f(t) is an odd function, then the function phi(x)=int_(a)^(x) f(t)dt

STATEMENT 1: If f(x) is continuous on [a,b] then there exists a point c in(a,b) such that int_(a)^(b)f(x)dx=f(x)(b-a) STATEMENT 2: For a