Definition clarification for Fourier transform

[space-time]

I have been studying Fourier transforms lately. Specifically, I have been studying the form of the formula that uses the square root of 2π in the definition. Now here is the problem:

In some sources, I see the forward and inverse transforms defined as such:
F(k) = [1/(√2π)] ∫^∞_-∞ f(x)e^ikx dx
f(x) = [1/(√2π)] ∫^∞_-∞ f(k)e^ikx dkIn other cases, I've seen:
F(k) = [1/(√2π)] ∫^∞_-∞ f(x)e^-ikx dx
f(x) = [1/(√2π)] ∫^∞_-∞ f(k)e^ikx dk

Notice that in the first version of the forward transform (the one that solves for F(k)), the exponential in the integrand has a positive sign in the exponent ikx, while in the 2nd version it has a negative ikx.

Which version is correct? Are they both correct and it is a matter of convention? Are neither correct?

Also, is there some way to do a multiple dimensional Fourier transform using volume integrals? If so, what is the formula for that (preferably including (√2π))?

 

[MisterX]

Only the 2nd pair is correct. There are a couple conventional issues, but no matter what the sign on the exponent has to change for the inverse transform relative to the forward transform.