Simplifying symbolic expression with Heaviside step function

Question

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?

Answer 1

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.