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\title{Math 1a.  Lecture 2 \\
Integration by  Substitution}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To be able to evaluate definite and indefinite integrals using substitution.


\end{itemize} 


\section{Integration by Substitution}

The general integration by substitution formula is
$$
\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du
$$
where $u = g(x)$ and $du = g'(x) \, dx$.

\section{Substitution for Definite Integrals}

If  $u = g(x)$ and $du = g'(x) \, dx$,
$$
\int_a^b  f(g(x)) \cdot g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du.
$$
Choose an antiderivative $F$ or $f$.  Then $F' = f$ and the Chain Rule implies
$$
\frac{d}{dx} F(g(x)) =  F'(g(x)) \cdot g'(x) = f(g(x)) \cdot g(x).
$$
Therefore,
\begin{eqnarray*}
\int f(g(x)) \cdot g'(x) \, dx 
& = &
F(g(x)) \Big]_a^b \\
& = &
F(g(b)) - F(g(a)) \\
& = &
F(u) \Big]_{g(a)}^{g(b)} \\
& = &
\int_{g(a)}^{g(b)} f(u) \, du.
\end{eqnarray*}

\section{Taking Advantage of Symmetry}

Suppose that $f$ is continuous on $[-a, a]$.  Then
\begin{itemize}

\item
If $f$ is even, then
$$
\int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx.
$$


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

\end{itemize}


\section{Key Examples}


\begin{itemize}

\begin{multicols}{2}

\item 
$\displaystyle \int (4x + 3)^{3} \, dx$



\item 
$\displaystyle \int x^3 \sqrt{9 - x^2} \, dx$

\item 
$\displaystyle \int_0^\pi 4 \sin^3 x \cos x \, dx$


\item 
$\displaystyle \int \tan x \, dx$



\end{multicols}

\item
Suppose that $\int_0^{12} g(x) \, dx = \pi$.  Evaluate $\int_0^4 g(3x) \, dx$.

\end{itemize}

\section*{References}

\begin{itemize}

\item
\S 5.6 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{February 10, 2006}



\end{document}

