I need to calculate the first and second moment of a log normal with mean=$\log S_0-(\mu-\frac{\sigma^2}{2})(t-t_0)$ and variance=$\sigma^2(t-t_0)$. I know that the answer should be $E[X]=S_0e^{\mu(t-t_0)}$. I fill in the equation. where I put $(t-t_0)=\Delta t$: $$\int_{0}^{\infty}Sf(S;t-t_0,S_0)dS=\int_{0}^{\infty}S\frac{1}{S\sigma \sqrt{2\pi\Delta t}}e^{-\frac{(\log S-\log S_0-(\mu-\frac{\sigma^2}{2})(\Delta t))^2}{2\sigma^2 \Delta t}}dS$$
Then I take $y=\log\frac{S}{S_0}$ $$\int_{0}^{\infty}\frac{1}{\sigma \sqrt{2\pi\Delta t}}e^{-\frac{(y-(\mu-\frac{\sigma^2}{2})(\Delta t))^2}{2\sigma^2 \Delta t}}dy$$
I think I need to make a quadratic part above the $e$ so I can make that integral 1 as it is a density function.
Can someone help me on how to do this?