What is the integral of (x*f(x))dx?

Answer 1

Do we have any idea what f(x) is?

I suppose in general you can integrate by parts

int u dv = u v - int[ v du]

here maybe u = f(x) and dv = x dx
du = f'(x) dx and v = x^2/2
then we would get
(x^2/2)f(x) - int [ (x^2/2) f'(x) dx]
but that seems rather pointless.

Answer 2

If you are asking what it is, consider this:

u=INT (x-c)^n * f(x) dx

is the definition of the nth moment about the mean c. so in math,
u=INT xf(x)dx is the first moment about the y axis (xo=c=0).
That is the definition, if that is what you were looking for. To calculate it, as Professor Damon stated, you have to know f(x).