Integrals That Can and Cannot Be Done
Evaluating integrals is much more difficult than evaluating derivatives. For derivatives, there is a systematic procedure based on the chain rule that effectively allows any derivative to be worked out. But for integrals, there is no such systematic procedure.
One of the main problems is that it is difficult to know what kinds of functions will be needed to evaluate a particular integral. When you work out a derivative, you always end up with functions that are of the same kind or simpler than the ones you started with. But when you work out integrals, you often end up needing to use functions that are much more complicated than the ones you started with.
The Wolfram Language includes a very wide range of mathematical functions, and by using these functions a great many integrals can be done. But it is still possible to find even fairly simple‐looking integrals that just cannot be done in terms of any standard mathematical functions.
The main point of being able to do an integral in terms of standard mathematical functions is that it lets one use the known properties of these functions to evaluate or manipulate the result one gets.
In the most convenient cases, integrals can be done purely in terms of elementary functions such as exponentials, logarithms and trigonometric functions. In fact, if you give an integrand that involves only such elementary functions, then one of the important capabilities of Integrate is that if the corresponding integral can be expressed in terms of elementary functions, then Integrate will essentially always succeed in finding it.
Beyond working with elementary functions, Integrate includes a large number of algorithms for dealing with special functions. Sometimes it uses a direct generalization of the procedure for elementary functions. But more often its strategy is first to try to write the integrand in a form that can be integrated in terms of certain sophisticated special functions, and then having done this to try to find reductions of these sophisticated functions to more familiar functions.
A large book of integral tables will list perhaps a few thousand indefinite integrals. The Wolfram Language can do essentially all of these integrals. And because it contains general algorithms rather than just specific cases, the Wolfram Language can actually do a vastly wider range of integrals.
Particularly if you introduce new mathematical functions of your own, you may want to teach the Wolfram Language new kinds of integrals. You can do this by making appropriate definitions for Integrate.
In the case of differentiation, the chain rule allows one to reduce all derivatives to a standard form, represented in the Wolfram Language using Derivative. But for integration, no such similar standard form exists, and as a result you often have to make definitions for several different versions of the same integral. Changes of variables and other transformations can rarely be done automatically by Integrate.