Wolfram Alpha: How to show functions with abs() and sgn() as a piecewise function?

Question

I was trying to obtain this result:

integrate e^|x| dx

The result is

$$ \int e^{|x|} dx = \frac12 e^{-x} \left( (e^x-1)^2 \operatorname{sgn}(x) -2e^x +e^{2x} - 1 \right) \color{gray}{\,+ \text{constant}} $$

which is fine but I would like to see the 2 part solution for $x>0$ and $x<0$, I know you can see that because just after before the final output I can see it but then it switches to the one line solution above. Is there a way to see the other solution?

If it's not ok to ask about a tool used in mathematics I'm sorry. You can close it.

Answer 1

Apparently, giving the input

Integrate[Exp[Piecewise[{{x, x > 0}}, -x]],x]

to WolframAlpha returns a piecewise result, which may be what you're expecting.

Answer 2

You can use the following to get the two cases:

Assuming[t>0, integrate e^|x| dx from x=0 to t]
Assuming[t<0, integrate e^|x| dx from x=0 to t]

Or, if you don't care about choosing the constant of integration such that the two pieces fit together, simply:

Assuming[x>0, integrate e^|x| dx]
Assuming[x<0, integrate e^|x| dx]

(although then it's so easy that I don't see why you would need Wolfram Alpha).