TITLE: Why isn't there a fixed procedure to find the integral of a function?
QUESTION [5 upvotes]: Since the integration of a function is the opposite of a the derivative of a function, and there are clear steps to follow when we want to find the derivative of a function, I thought there would be clear steps to follow too when we want to find the integral of a function.
What is it about finding the integration of a function that makes it impossible to come up with a guaranteed method to find it?
Edit:
I am only considering elementary functions.

REPLY [6 votes]: Because the class of function you consider easy to differentiate—namely, the elementary functions—is not mapped onto itself.  You are accustomed to thinking of elementary functions as being exactly those on which you "do calculus" because they are easy to differentiate, but a randomly chosen one may not be in the image of the derivative function, and therefore, not easy to integrate.  The standard example $f(x) = e^{x^2}$ applies here; no simple example of a hard-to-differentiate function exists, by definition.
Note that without the assurance that functions are given by explicit formulas of a convenient form, neither integration nor differentiation is particularly easy.