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Why does the behave this particular way:

    WolframAlpha["integrate integrate x times y from x=0 to 1 and from y=-1 to 1", IncludePods -> "Input", AppearanceElements -> {"Pods"}, 
 TimeConstraint -> {60, Automatic, Automatic, Automatic}, 
 PodStates -> {"Input__Step-by-step solution"}]

The solution is correct. Since the y-integration is symmetric by the interval and the integrant is antisymmetric, the solution is zero.

Problem remains, no steps given by the WolframAlpha function.

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    $\begingroup$ Bivariate definite integration show-steps might not be a supported pod state. $\endgroup$ – Daniel Lichtblau Jan 13 at 22:07
  • $\begingroup$ @DanielLichtblau This is correct. Alpha only has steps for univariate integrals. $\endgroup$ – Chip Hurst Jan 15 at 17:58
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You can split it into two steps:

WolframAlpha[" integrate x times y from x=0 to 1 ", IncludePods -> "Input", 
AppearanceElements -> {"Pods"}, TimeConstraint -> {60, Automatic, Automatic, Automatic}, 
PodStates -> {"Input__Step-by-step solution"}]

and

WolframAlpha["integrate y/2 from y=-1 to 1", IncludePods -> "Input", 
 AppearanceElements -> {"Pods"},  TimeConstraint -> {60, Automatic, Automatic, Automatic}, 
PodStates -> {"Input__Step-by-step solution"}] 

Perhaps, the final result of the first step, i.e. y/2, can be entered in the second step programmatically.

PS. I am aware of "So questions about WolframAlpha[] and related Mathematica functions are ok, but not questions such as this one seems to be about the workings of W|A". However, the use of W|A in Mathematica is the only known to me way to obtain step-by-step solutions in Mathematica so I posted this answer.

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