I want to calculate the probability distribution from the characteristic function given as
$$G(k) = e^{ik\mu_1 - \frac{\sigma^2 k^2}{2}}$$
The probability distribution is given as inverse Fourier transform of the characteristic function
$$P(x) = \int \frac{dk}{2\pi} e^{-ikx} G(x)$$
I'm stuck in solving this integral. The integral of this results in the Gaussian distribution. Can someone help me out solving this integral?