I'm learning about the Fourier transform and have learned that the Fourier transform is $$\hat{f}(\omega)=\int_{-\infty}^{\infty}f(x)e^{-2i\pi\omega x}dx$$ while the inverse transform is $$f(x)=\int_{-\infty}^{\infty}\hat{f}(\omega)e^{2i\pi\omega x}d\omega$$
Why is the inverse Fourier transform another integral? Isn't the opposite operation of an integral a derivative?