Given a function `f(x)` such that It is integrable over every interval on the real line, and `f(t+x)=f(x),` for every `x` and a real `tdot` Then show that the integral `int_a^(a+t)f(x)dx` is independent of `adot`

If f(x) is an integrable function over every interval on the real line such that f(t+x)=f(x) for every x and real t, then int_(a)^(a+t) f(x)dx is equal to

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

The value of the integral int_(0)^(2a) (f(x))/(f(x)+f(2a-x))dx is equal to

If f(x)=x+int_0^1 t(x+t) f(t)dt, then find the value of the definite integral int_0^1 f(x)dx.

The integral int_(0)^(a) (g(x))/(f(x)+f(a-x))dx vanishes, if

Let f(x) = x-[x] , for every real number x, where [x] is integral part of x. Then int_(-1) ^1 f(x) dx is

Integrate the functions f(x) f(x)=x log(x)

Evaluate each of the following integral: int_a^b(f(x))/(f(x)+f(a+b-x))dx

If f(1 + x) = f(1 - x) (AA x in R) , then the value of the integral I = int_(-7)^(9)(f(x))/(f(x)+f(2-x))dx is

A function f, continuous on the positive real axis, has the property that for all choices of x gt 0 and y gt 0, the integral int_(x)^(xy)f(t)dt is independent of x (and therefore depends only on y). If f(2) = 2, then int_(1)^(e)f(t)dt is equal to