I'm trying to understand how to derive the cumulative distribution function for a lognormal distribution from its probability density function.

I know that the pdf is:

$$f(x) =\frac{e^{-\frac{1}{2}\Bigl(\frac{\ln(x) - \mu}{\sigma}\Bigr)^2}}{x \sigma \sqrt{2\pi}},\ x \gt 0$$

and the cdf is:

$$\Phi(x) = \int\_{-\infty}^x f(y) dy = \frac{1}{\sigma\sqrt{2\pi}}\int\_0^x e^{-\frac{1}{2}\Bigl(\frac{\ln(y) - \mu}{\sigma}\Bigr)^2}\frac{dy}{y}.$$

Now, I don't know how to get the last formula. I tried the substitution $t=\ln(y)$ but I don't know how to deal with the $1/y$ term.

Am I on the wrong way, or did I make any mistake?

I'd like to know it for my personal knowledge, it's not for a class.