your thinking won't be true in general (and we don't even have a definite integral here). Take , for instance. I think we need more info here, if you have some definite answer to get to, as opposed to an expression.
x(t) = u(t) - u(t-10)
Am I thinking about this correctly... u(t) is 0 from neg. infinity to 0, and 1 after. u(t-10) is the same thing just shifted to the right to "turn on" at 10. So in effect we have a rectangle of amplitude 1 with bounds from 0 to 10. So our integral becomes:
int (1)^2 dt (with bounds of 0 and 10), which = 10?
I'm assuming that is intended to notate the Heaviside step function, not an arbitrary function.
In any case, could you clarify whether this is a definite or indefinite integral? If we take it as written, it would be an indefinite integral, i.e.,
As you observed, you could define as a piecewise function like so:
Then we could find the anti-derivative using piecewise techniques.
If instead you are calculating a definite integral whose bounds cover the 0 to 10 range, then your work is correct.