The Chain Rule is a way of finding derivatives by substitutions. Integration by substitution is the Chain Rule for finding antiderivatives. That is it uses substitutions to help one to recognizes the indefinite integral of a function.
To facilitate the discussion we introduce differential forms . An in-depth treatment of the subject can be found in [2]. We think of differential forms, instead of functions, as formal objects for integration. For instance, instead of integrating the function \(x\sin(x^2)\text{,}\) we think of integrating the form \(x\sin(x^2) dx\text{.}\)
A differential 1-form, or simply a differential form (in \(x\)), is an expression of the form \(f(x) dx\text{.}\) Given a differentiable function \(f(x)\text{,}\) we write \(df(x)\) (or simply \(df\)) for the differential form \(f'(x)\ dx\text{.}\) We say that a differential form is exact if it is \(df(x)\) for some differentiable function \(f(x)\text{.}\) Integrating an exact form is trivial because by definition \(f(x)\) is an antiderivative of \(f'(x)\) and so,
To integrate by substitution means making various substitutions until the differential form to be integrated is transformed into a differential form that one recognizes as exact.
Example2.4.
To integrate \(x\sin(x^2)\ dx\text{,}\) first realize that integrating \(\sin(x^2) dx\) is a problem but integrating \(\sin(u) du\) is not. This suggests the substitution \(u = x^2\text{.}\) With that we get \(du = dx^2 = 2xdx\text{.}\) Thus,
\begin{equation*}
x\sin(x^2)\ dx = \frac{1}{2}\sin(u)\ du = -\frac{1}{2}d\cos(u)
\end{equation*}
and so
\begin{align*}
\int x\sin(x^2)\ dx & = \int \frac{1}{2}\sin(u) du \\
&= -\frac{1}{2}\cos(u) +C = -\frac{1}{2}\sin(x^2) + C
\end{align*}
Example2.5.
To compute the indefinite integral \(\int \tan(x) dx\text{,}\) observe that \(\sin(x) dx = -d \cos(x)\text{.}\) So, let \(u=\cos(x)\)
