You are right, however the same way with automatic differentiation, you never get the expression, with automatic integration it would be the same.
When you input the variable values into a symbolic derivative you just get a value at the end. d/dx x^2 = 2x. If x = 0.5 then d/dx = 1. The same is true for symbolic integrals. For most practical applications, we don't really care about the full symbolic expression. We just want the answer, or at least a good approximation. This post uses a specific example of the difference between two Beta distributions. We want to get that 0.71. It is very hard to "automatically" make that happen.
But what about something like cos(x). Either you can lazily evaluate the power series or you know its sin(x).
Say we want the derivative of f(x) = x^3 - 2x^2 + 5. That becomes:
(x+e)^3 - 2(x+e)^2 + 5
= x^3 + 3x^2e - 2x^2 -4xe + 5
= (x^3 - 2x^2 + 5) + (3x^2 - 4x)e
The term in front of 'e' is "3x^2 - 4x", which is f'(x).
f'(x) = lim_{e->0} (f(x+e) - f(x))/e
If you're able to express f(x+e) on the form (f(x) + y e) then it follows that y is the derivative.
It also should be noted that auto-diff doesn't let you skip the rules for derivation and you're using the same calculation as you would to show that e.g. f'(x^n)=n*x^{n-1}.
What's the integral of exp( sqrt ( 1 + (tan^(3/2) X)2 ) ) ) ?
We only know a handful of forms that can be integrated in closed form and its down to our creativity to discover new forms that can be integrated (same deal with solving differential equations and the reasons are the same).
The forms that we know how to integrate can be done by a computer. CAS tools will do that for you. For example Mathematica.
Finding an "auto-integrate method" would probably involve finding a way of calculating the integral in a decomposable way, and that indeed would be amazing, but I don't really see that happening any time soon.