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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/The Substitution Rule">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 21:47:08 1997}
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\section{The Substitution Rule\QTR{BOXHEAD}{\label{substitution rule}}}

\subsubsection{\QTR{BOXHEAD}{The Substitution Rule}}

If $u=g(x)$ is a differentiable function whose range is an interval $I$ and $%
f$ is continuous on $I$, then 
\[
\int f(g(x))g^{\prime }(x)\,dx=\int f(u)\,du 
\]

\subsubsection{\QTR{BOXHEAD}{The Substitution Rule for Definite Integrals%
\label{substitution rule for definite integrals}}}

If $g^{\prime }$ is continuous on $[a,b]$ and $f$ is continuous on the range
of $g$, then 
\[
\int_{a}^{b}\,\,f(g(x))g^{\prime }(x)\,dx=\int_{g(a)}^{g(b)}\,\,f(u)\,du 
\]

\subsubsection{\QTR{BOXHEAD}{Integrals of Symmetric Functions\label%
{integrals of symmetric functions}}}

Suppose $f$ is continuous on $[-a,a]$.

\begin{enumerate}
\item  If $f$ is even $\lbrack f(-x)=f(x)\rbrack $, then $%
\int_{-a}^{a}f(x)\,dx=2\int_{0}^{a}f(x)\,dx.$

\item  If $f$ is odd $\lbrack f(-x)=-f(x)\rbrack $, then $%
\int_{-a}^{a}f(x)\,dx=0.$
\end{enumerate}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Calculus, change variable}{}{}{../../Help/DM7-5.tex#Change
of variables}}

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