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

I was trying to obtain this result:

integrate e^|x| dx


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

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.

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Well, you can derive it yourself; $\mathrm{sgn}(x)$ is expressible as a piecewise function. –  Ｊ. Ｍ. Sep 3 '10 at 20:10
I cannot see what you are asking here... «The other solution» is the same as the solution you have. –  Mariano Suárez-Alvarez Sep 6 '10 at 15:07
btw, Wolfram Alpha uses Mathematica as a backend, you may need to use Mathematica syntax to get more advanced things out of it –  Yaroslav Bulatov Sep 6 '10 at 16:35

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.

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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).

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I'm playing with the tool :D I was testing something "complicated" from the point of view of the tool, not the math, to see how flexible it is. –  dierre Sep 4 '10 at 8:06
BTW nice workaround but since I see the solution I want just before it switches on the one line solution, I would like to see the 2part solution without giving it a restraint on the variable. –  dierre Sep 4 '10 at 8:11
What do you mean by "seeing the solution"? For me, Alpha just spits out the answer that you quoted in the question. (No "switching" going on.) –  Hans Lundmark Sep 4 '10 at 9:44
There's a tiny moment, it's a one second switch. Clear your cache, maybe that's the problem. Or try with integrate e^-|x| dx  which has a slower process. –  dierre Sep 4 '10 at 9:59
Nope, didn't make any difference. Well, never mind... –  Hans Lundmark Sep 4 '10 at 12:40