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\title{Math 1b. Trigonometric Integrals and Trigonometric Substitution}

\date{Spring 2006}

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\maketitle


\subsection*{Some Useful Identities}

\begin{gather*}
\cos^2 x + \sin^2 x =1 \qquad
\sec^2 x = \tan^2 x + 1 \\
\sin^2 x = \frac{1}{2}(1 - \cos 2x) \qquad
\cos^2 x = \frac{1}{2}(1 + \cos 2x)
\end{gather*}


\subsection*{Worksheet Exercises}

Evaluate each of the following integrals
\begin{enumerate}

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

\vfill



\item 
$\displaystyle \int \sin^2 x \cos^2 x \, dx$

\vfill

\clearpage

\item 
$\displaystyle \int \sec^3 x \tan^3 x \, dx$.

\vfill

\item 
$\displaystyle \int \sec \theta \, d\theta =  \int \frac{\cos \theta}{\cos^2 \theta} \, d\theta =  \int \frac{\cos \theta}{1 - \sin^2 \theta} \, d\theta $.  {\em Hint}: Make use of the identity
$$
\frac{2}{1 - u^2} = \frac{1}{1 - u} + \frac{1}{1+ u}.
$$

\vfill



\item 
$\displaystyle \int \sqrt{4 - x^2} \, dx$


\vfill

\item 
$\displaystyle \int \frac{1}{\sqrt{4 + x^2}} \, dx$


\vfill


\end{enumerate}



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