How to solve the integral $\int \frac{1}{e^x}\,dx$ step by step?

How to solve the integral $\int \frac{1}{e^x}\,dx$ step by step?

I'm primitive in integrals and derivatives and I'm trying to solve the
integral $\int \frac{1}{e^x}\,dx$, but especially this integral was hard
to me to solve it.
So I tried:
$$\begin{align} \int\frac{1}{e^x}\,dx&=\int
\frac{1}{\color{Red}{e^x}}\color{Blue}{e^x\,dx}\\&=\int
\frac{1}{\color{Red}{u}}\,\color{Blue}{du}\\&=\ln\left(|u|\right)\\&=\ln
\left(|e^x|\right)+C \end{align}$$
But my solution is wrong while I used the integration by substitution
method ?!



Correct answer:
$$\begin{align} \int\frac{1}{e^x}\,dx&=\left(-e^{-x}\right)+C \end{align}$$