Why do people say "find the integral" instead of "evaluate the integral"?

Solution 1:

To take a trivial example, consider the function f(x)=x.

We may “find” (in the sense of determining its form) the integral function of f as I(x)=∫x.dx = x.x/2. Thus far we have only determined (found) the integral function; it has not been evaluated (in the sense of given a numerical value). That can only be done once we know the limits of x between which the integral function I(x) is to be calculated.

Hence, the integral we found may evaluate to 2 if the limits of x are (0,2), or to 6 if the limits are (2,4).

Finding is therefore a separate thing to evaluating.

Despite having no comma, your example “find the integral ∫ f(x) dx” may reasonably be understood in mathematical prose as appositional: “find the integral, ∫ f(x) dx.