I have seen many solutions of integration of absolute values such as $\int |x| dx$ however how do you proceed when you have something like this $\int x |dx|$. I have been struggling with this for quite some time:
The problem:
I want to postulate a decay exponential functions such: \begin{gather} y = a (1- \exp(-bx))\\ \dot{y} = b (a - y) dx \end{gather} where $a$ is the maximum value that $y$ can have and $b$ is just the rate at which this value will be reached. However, when $x < 0 $ $y$ will tend to $\inf$. The next thing I did was to postulate the following rate
\begin{align} \dot{y} = b (a - y) |dx| \end{align} which will plot a decay response for $x>0$ and $x<0$, which it is what I want. The question is: how to integrate this last equation? Thanks!