Section 5.7 Substitution
The formulas in differential form given at the end of Section 5.5 not only summarize the basic differentiation formulas, but also the basic integration formulas! In each case, one can integrate both sides of the expression, then use the Fundamental Theorem of Calculus, which says that
which can be interpreted as a statement about antiderivatives. For example, knowing that
immediately tells us that
which is normally rewritten in the form
where \(m=n+1\text{.}\)
Some of the differentiation rules introduced in Section 5.6 also lead to integration rules. For example, the product rule in the form
immediately yields
which is normally rewritten in the form
where it is known as integration by parts.
But the most important of these integration rules is the analog of the chain rule. Multiplying both sides of
by an arbitrary function \(f\) and then integrating yields first
and then
which is the technique known as substitution. 1 For example, to evaluate the integral
it is enough to notice that \(d\sin\theta=\cos\theta\,d\theta\text{,}\) so that the integral becomes
where the substitution \(u=\sin\theta\) has been made implicitly.