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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Rationalizing Substitutions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Rationalizing Substitutions\label{rationalizing substitutions}}

Some functions can be changed into rational functions and integrated by the
method of \hyperref{partial fractions}{}{}{cal07_04.tex#partial fractions}
by making appropriate substitutions.

\begin{itemize}
\item  When an integrand contains an expression of the form $\sqrt[n]{%
g\left( x\right) }$, the substitution $u=\sqrt[n]{g\left( x\right) }$ may be
useful.

\item  The substitution $t=\tan \left( \frac{x}{2}\right) $ will convert any
function of $\sin x$ and $\cos x$ into a rational function of $t$. Let 
\[
\begin{array}{ll}
t=\tan \left( \frac{x}{2}\right) \text{ } & -\pi <x<\pi 
\end{array}
\]
Then 
\[
\begin{array}{l}
\sin x=\frac{2t}{1+t^{2}} \\ 
\\ 
\cos x=\frac{1-t^{2}}{1+t^{2}} \\ 
\\ 
dx=\frac{2}{1+t^{2}}dt
\end{array}
\]
\end{itemize}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Changing variables}{}{}{../../Help/DM7-5.tex#Change of
variables}}

{\small \hyperref{Factoring polynomials}{}{}{%
../../Help/DM3-1.tex#Polynomials, factor}}

{\small \hyperref{Method of partial fractions}{}{}{%
../../Help/DM7-5.tex#Partial fractions}}

{\small \hyperref{Partial fractions}{}{}{../../Help/DM3-1.tex#Partial
fractions}}

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