Why do we drop the abolute value bars when doing indefinite integration? Say, for instance, we're trying to solve $\frac{dy}{dx}=y\cos(x)$.
Separating the variables, we get: $$\int \frac{1}{y}dy=\int \cos(x) dx.$$
So $\ln|y|=\sin(x)+c \iff |y|=Ae^{\sin(x)}$ (where $A:=e^c$), yielding:
$$y=\pm Ae^{\sin(x)}$$, which is inconsistent with Wolfram Alpha's $y=Ae^{\sin(x)}$.
Now, I know that $A$ could be either positive or negative, but it's a constant, so it's either one or the other.
So, what I'm asking is: how can WA justify dropping the absolute value bars?
Thanks

$-Ae^{\sin x}$ is just $Ae^{\sin x}$ with a different $A$.
