stochastic calculus - Close form solution for Geometric Brownian Motion

I have a very fundamental problem, please help me out. I am little confused with the derivation for the close form solution for the Geometric Brownian Motion, from the very fundamental stock model: $$\begin{equation} dS(t)=\mu S(t)dt+\sigma S(t)dW(t) \end{equation} $$ The close form of the above model is following: $$ \begin{equation} S(T)=S(t)\exp((\mu-\frac1 2\sigma^2)(T-t)+\sigma(W(T)-W(t))) \end{equation} $$

I believe this is quite straightforward for most of you guys, but I really dont know how did you get the $(\mu-\frac 1 2 \sigma^2)$ term. It is clear for me the other way round (from bottom to top), but I fail to derive directly from the top to bottom. I checked some material online, it was saying something with the drift term, which some terms are artificially added during the derivation.

Your answer and detailed explanation will be greatly appreciated.

Thanks in advance!