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When inputing

Integrate[HeavisideTheta[s - t], {s, 0, 1}, Assumptions -> {t > 0}]

Mathematica correctly simplifies if to

-(-1 + t) HeavisideTheta[1 - t]

even though I did not explicitly ask it to Simplify. However, if I include an undefined function of the variable integrated over, x[s], Mathematica returns a wrong (or, rather, incomplete) result:

Integrate[HeavisideTheta[s - t] x[s], {s, 0, 1},  Assumptions -> {t > 0}]

ConditionalExpression[0, t > 1]

Is there any way to fix this, e.g. leaving the expression unsimplified in the cases where indeed no simplifications can be made a priori?

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  • $\begingroup$ Can you explain what answer you expect to see? $\endgroup$
    – bill s
    Commented Aug 22, 2019 at 3:26
  • $\begingroup$ It would be enough to leave it unevaluated, in such a way that if I then assign the function x[s] and Simplify I get the correct result. $\endgroup$
    – sdnnds
    Commented Aug 22, 2019 at 12:20

1 Answer 1

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One approach is to define your integral as a function, which you can then call with your desired x[s]:

myHeavy[x_] := Integrate[HeavisideTheta[s - t] x, {s, 0, 1}, Assumptions -> {t > 0}];

Now, when you want to call this with a function x[s] = s^2, you evaluate

myHeavy[s^2]

To evaluate with Sin[s]

myHeavy[Sin[s]]

These seem to return plausible answers.

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