If you are thinking of the DFT as an approximation for a continuous Fourier transform, then you should multiply it by Δω in order for the sum to be an approximation for \int d\omega. (The finite window is also an approximation.)
This has nothing to do with Julia or the FFTW.jl package. Any FFT function computes the discrete Fourier transform (DFT), not the continuous Fourier transform, and you need to understand how the two relate for your application.