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Section 7.7 Substitution

The formulas in differential form given at the end of Section 7.5 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

\begin{equation} \int df = f\tag{7.7.1} \end{equation}

which can be interpreted as a statement about antiderivatives. For example, knowing that

\begin{equation} d(u^n) = nu^{n-1}\,du\tag{7.7.2} \end{equation}

immediately tells us that

\begin{equation} u^n = \int d(u^n) = \int nu^{n-1}\,du\tag{7.7.3} \end{equation}

which is normally rewritten in the form

\begin{equation} \int u^m\,du = \frac{u^m}{m+1}\tag{7.7.4} \end{equation}

where \(m=n+1\text{.}\)

Some of the differentiation rules introduced in Section 7.6 also lead to integration rules. For example, the product rule in the form

\begin{equation} d(uv) = u\,dv + v\,du\tag{7.7.5} \end{equation}

immediately yields

\begin{equation} uv = \int d(uv) = \int u\,dv + \int v\,du\tag{7.7.6} \end{equation}

which is normally rewritten in the form

\begin{equation} \int v\,du = uv - \int u\,dv\tag{7.7.7} \end{equation}

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

\begin{equation} du = \frac{du}{dx}\,dx\tag{7.7.8} \end{equation}

by an arbitrary function \(f\) and then integrating yields first

\begin{equation} f\,du = f\,\frac{du}{dx}\,dx\tag{7.7.9} \end{equation}

and then

\begin{equation} \int f\,du = \int f\,\frac{du}{dx}\,dx\tag{7.7.10} \end{equation}

which is the technique known as substitution.  1  For example, to evaluate the integral

\begin{equation} Q = \int \sin^2\theta\cos\theta\,d\theta\tag{7.7.11} \end{equation}

it is enough to notice that \(d\sin\theta=\cos\theta\,d\theta\text{,}\) so that the integral becomes

\begin{equation} Q = \int \sin^2\theta\,d(\sin\theta) = \frac13 \sin^3\theta\tag{7.7.12} \end{equation}

where the substitution \(u=\sin\theta\) has been made implicitly.

This technique is often referred to as “\(u\)-substitution” even when there is no variable named \(u\text{.}\)