Trying to understand why the signal was inverted on this RC step response equation
It's the same either way. You don't have to multiple by -1. Try it:

$$\begin{align}
\frac{dt}{RC} &amp;=\frac{dv}{v_s-v}\\
\\
\int_{v_0}^v \frac{dv}{v_s-v} &amp;= \int_{0}^t \frac{dt}{RC}\\
\end{align}$$


let's forget about the right side, since it doesn't pertain to your question.

Using

$$\begin{align}
\int \frac{1}{ax+b}dx = \frac{1}{a}ln|ax+b|\\
\end{align}$$

then \$a=-1\$ and \$b=v_s\$ so

$$\begin{align}
\int_{v_0}^v \frac{dv}{v_s-v} \rightarrow(-1)ln|v_s-v|\\
\end{align}$$

now follow the rest of the tutorial and you'll eventually end up with

$$\begin{align}
-v &amp;= -v_s + (v_s-v_0)e^{\frac{-t}{RC}}\\
\end{align}$$

at which point, obviously, to solve for \$v\$, multiply both sides by -1.