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{\Large Trigonometric Substitution

February 5, 2003}
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The point of trigonometric substitutions are to handle integrals with an integrand that contains expressions of the form:
\[
\sqrt{a^2-x^2} \qquad \sqrt{a^2+x^2} \qquad \sqrt{x^2-a^2}
\]

To handle these we make a substitution to take advantage of trigonometric identities.
You can also make \emph{hyperbolic} trigonometric substitutions which take advantage of 
the hyperbolic trigonometric identities.  
(You will not be required to know the hyperbolic trigonometric substitutions, 
but I thought you might be interested in knowing that it is possible.)

Here is a table of when to use what substitution and what trigonometric substitution to use:

\[
\begin{array}{|r|r|r|}
\hline
\mbox{Integrand} & \mbox{Trig Identity} & \mbox{Substitution} \\
\hline
\sqrt{a^2-x^2} & 1-\sin^2\theta = \cos^2\theta & x=a\sin\theta \\
\sqrt{a^2+x^2} & 1+\tan^2\theta = \sec^2 \theta & x=a\tan\theta \\
\sqrt{x^2-a^2} & \sec^2\theta-1  = \tan^2 \theta & x=a\sec\theta \\
\hline
\end{array}
\]

Here is the general strategy:
\begin{enumerate}
\item  Determine which substitution to make.
\item  Make the substitution (and convert the integral to an integral in $\theta$).
\item  Integrate (anti differentiate).
\item  Convert back to $x$'s.  
\end{enumerate}

We will work some of these through in class.

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