Difference between Forward and Backward Fourier Transforms?

What is the difference between forward and backward Fourier transforms? I'm look:

$$F(k) = \int_{-\infty}^{\infty} f(x)\ e^{- i 2\pi k x }\,dx$$

$$f(x) = \int_{-\infty}^{\infty} F(k)\ e^{ i 2\pi k x }\,dk$$

If I swap the x and the k in the second equation, the transforms are then:

$$F(k) = \int_{-\infty}^{\infty} f(x)\ e^{- i 2\pi k x }\,dx$$

$$F(k) = \int_{-\infty}^{\infty} f(x)\ e^{ i 2\pi x k }\,dx$$

and the only difference is the minus sign in the exponent. What gives? Why aren't the forward and backwards transforms identical?