Proof of expected value E[exp(x)] = exp(mean + 0.5 * variance) for normally distributed variable x

Below is the link of the proof of integral of the probability density function of the standard normal distribution.

In the video, myu = the mean or expected value of x, i.e. E[x], sigma = the standard deviation of x. This equation is one of the most important properties in the financial maths and one of the key concepts of the famous Black-Scholes-Merton formula. If you set Y = exp(x), ln(Y) is normally distributed and Y is log-normally distributed. I know it sounds confusing.