19.3 Anti-derivatives by Inspection and Classes of Functions that we can Anti-differentiate

The process of differentiating a known standard function can be achieved by parsing the function definition, and applying the appropriate rules of differentiation to each step in its definition.

There are no specific anti-differentiation rules; we have to use the differentiation rules backwards. And these allow us to anti-differentiate large classes of standard functions, but not all of them.

We can recognize some functions almost immediately as derivatives of others. For example, we can anti-differentiate any polynomial in x by applying the rule to differentiate powers backwards.

Thus an anti-derivative of (When anti-differentiating polynomials, a standard error is to suffer a mental lapse in the middle of doing this and differentiate instead of anti-differentiating some term; always check that you haven't done this.)

We can similarly anti-differentiate any power (even , one of whose anti-derivatives is ln(x)). And we can anti-differentiate e^ax and any polynomial in sin(x) and cos(x), although doing the latter requires applying some tricks.

It turns out, that there are more tricks which allow us to differentiate any rational function where p and q are polynomials, if we can factor q(x) into linear and quadratic factors; and tricks which allow us anti-differentiate an exponential times a polynomial, inverse powers of sines and cosines, functions that are polynomials divided by the square root of a linear or quadratic functions, products of exponentials sines and polynomials, and various other classes of functions.

On the other hand there are simple looking standard functions that we cannot anti-differentiate, because their anti-derivatives do not happen to be standard functions.

(If these anti-derivatives were sufficiently important we would give them names, make tables of them and add them to the list of our basic functions (which consists now of x, sin x and e^x). As it happens they are important but not that important; we give them names and tabulate them, but call them special functions.

The simplest functions that cannot be anti-differentiated as standard functions are both of which have special functions as specific anti-derivatives.)

There used to be a difference of opinion among teachers of calculus as to whether students should be made to learn all the tricks of anti-differentiation so that they could anti-differentiate anything that could be anti-differentiated.

Some felt that the mark of mastery of calculus was to be able to anti-differentiate strange looking functions by attempting to apply the various tricks. Others pointed out that there are "integral tables" which list essentially all anti-differentiable functions and their anti-derivatives, so that this skill, whose development is a wonderful exercise of ingenuity and memory, has very limited practical value.

At that time there were obstacles that students faced in getting a copy of a reasonably complete integral table, and in looking things up in huge ones, that gave the edge by a small margin to the first viewpoint.

Today, however, there are commercially available programs, Maple, MATLAB, and Mathematica (and perhaps others), which give you detailed formulae for the anti-derivative of any function you care to enter, and do so instantly; which seems to tilt the argument toward the second viewpoint.

Be that as it may, every student of calculus should be aware of the four basic tools of anti-differentiation, which we now discuss.

Before doing so we should notice the important fact that the action of finding an anti-derivative is a linear operation (as differentiating is) so that an anti-derivative of a sum is the sum of anti-derivatives of its summands.