I have the following simple multiplication of integral by itself:


I feel that I probably miss something very basic, but is it possible to write it in some form like:


I faced this integral in integral equation and was surprised when didn't find literature handling this, so I guess easy transformation must exists.

$a(t)$ is a very general differentiable function.


First of all you are abusing the notation, since the variable you are integrating over and the limits are the same variable $t$.

And to address your question (assuming your notation is fixed), yes, you can where $$f(a(t),t) = a(t) \int_0^x a(y)dy$$

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