Section Summary

Evaluating indefinite integrals, i.e., finding antiderivatives, can be a process of guessing and correcting - trial and error. Often it is not evident how to start the process, how to make the first guess. However, substitution can change a difficult problem into a more tractable one. This method is simply the Chain Rule in reverse. If you can rewrite the integral in question in the form \(\displaystyle\int\,f(u(x))u'(x)\,dx\), i.e., as \(\displaystyle\int\,f(u)\,du\), then you only need to find an antiderivative \(F\) for \(f\). In that case,

There is a corresponding result for definite integrals:

We referred back to our work in Chapter 5 on the inverse trigonometric functions (in particular, arctangent and arcsine) and their derivatives, which turn out to be useful for integrating functions that involve sums and differences of squares. Specifically, if the integrand has a factor of \(a^2+x^2\) (where \(a\) is a constant), then an arctangent substitution is likely to be useful, and if the integrand involves \(a^2-x^2\), then an arcsine substitution may transform the integral to a familiar form.

Also in this section we have seen that the Product Rule for differentiation,

(in differential form), leads to the integration-by-parts formula,

This technique allows you to replace one integration problem with another. It is useful if the new problem is simpler than the original one. Thus, to apply this formula effectively, you need to consider what happens to each factor when you differentiate one and integrate the other. If it is not clear whether \(\displaystyle\int\,v\,du\) is simpler than \(\displaystyle\int\,u\,dv\), then you may have to try more than one factorization to find one that works.