Section Summary

We have explored in this section the benefits and risks of using a computer algebra system, in particular, Sage, to evaluate indefinite integrals, i.e., to find antiderivatives. Throughout we have emphasized the importance of a human-machine partnership, of combining what you know about algebra with what the CAS can reasonably be expected to do.

In Section 8.4 you learned several ways to transform an integral problem to some other integral problem: algebraic substitutions, trigonometric substitutions, and integration by parts. At the start of this section, in the context of integrating rational functions, we made use of another transformation scheme, that of rewriting the integrand in another algebraic form. When your CAS won't solve the problem in the form you enter it, it's your job to think of a useful transformation that will make the problem more tractable for the CAS.

We have also emphasized the importance of checking antidifferentiation by differentiating the CAS answer, an easy step for the CAS to do, and with 100% accuracy. (In particular, we have built this step into our Indefinite Integral tool.) One reason for checking is that it's very easy to make a typing mistake or an algebraic grouping mistake in telling the CAS what you want to integrate. And if you don't know what the answer should look like, you may not pick up that mistake just by looking at the answer. In general, checking should always be a part of your mathematical strategy, but especially when the problem at hand is hard and the checking mechanism is easy.

Finally, we observed the usefulness of other tools built into computer algebra systems, especially graphing the difference of two functions. When we want to know whether two algebraic expressions are equivalent — or differ by a constant — we can ask the Graph tool to plot their difference.