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I am trying to use Wolfram Alpha in Mathematica 10.2 to show me the steps for calculating the integral in the title, i.e I have written the following command:

show steps of Integrate[Exp[-(I*(s*t))]/(Sqrt[2*Pi]*Sqrt[s]), {s, 0, Infinity}]

in the Wolfram alpha command line, but it doesn't show me the steps, can anyone help me on how to make Mathematica show me the steps for calculating this integral?

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    $\begingroup$ Mathematica and more in general computer algebra systems do not carry out integration in the same way one would in basic calculus, so the steps would not be either recognizable or particularly useful. This topic has been discussed before: Is there a way for me to get Mathematica show its steps? $\endgroup$ – MarcoB Dec 6 '15 at 20:00
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    $\begingroup$ See also Get a "step-by-step" evaluation in Mathematica $\endgroup$ – MarcoB Dec 6 '15 at 20:02
  • $\begingroup$ This answer shows how, but I suspect the reason you're not seeing the step-by-step button is because Mathematica / W|A does not know how to break it down. (Generally the steps are not the steps Mathematica takes.) This answer shows how to get an outline of the steps Integrate takes. $\endgroup$ – Michael E2 Dec 6 '15 at 20:06
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    $\begingroup$ Step-by-step integrals in alpha only are currently available for integrands with elementary anti derivatives, and this one is not elementary (it's in terms of Erf): In[1]:= Integrate[Exp[-(I*(s*t))]/(Sqrt[2*Pi]*Sqrt[s]), s] Out[1]= -(((-1)^(1/4) Erfi[(-1)^(3/4) Sqrt[s] Sqrt[t]])/( Sqrt[2] Sqrt[t])) $\endgroup$ – Chip Hurst Dec 6 '15 at 20:17
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    $\begingroup$ I'm voting to close this question as off-topic because Wolfram|Alpha questions are out of scope for Mathematica.StackExchange.com $\endgroup$ – Daniel Lichtblau Dec 6 '15 at 23:36
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Rubi integration package by Albert Rich, shows the steps, using its rules, for indefinite integration.

I've used this to show the steps for each of the over 51,000 integrals in Albert's independent integration test suite.

Here is how you'd do the above for this example

After you load the package, then type

stepInt[Exp[-(I*(s*t))]/(Sqrt[2*Pi]*Sqrt[s]), s]

It will now show that it used 3 rules to get to the final result

Mathematica graphics

The stepInt is thanks to Albert. Here it is. It will continue printing each step until it reaches the final result, or until it determines it can't solve it.

stepInt[u_, x_Symbol] := 
 Block[{ShowSteps = True}, FixedPoint[Function[Print[#];
    ReplaceAll[#, {Defer[Int] -> Int, Defer[Dist] -> Dist, 
      Defer[Subst] -> Subst}]], Int[u, x]];
  Null]
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