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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Practice Day 2 on Trigonometric Integrals}
\centerline{March 13, 2018}
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\noindent
{\it Background}
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Our approach to trigonometric integrals will be based in systematic
application of \emph{reduction formulas}.  When you do these on quizzes
and exams, formulas such as the following \emph{will be provided} for your
use in table format.  You will be responsible for deciding which formula
applies and for applying it correctly.  Here is the second ``batch of'' these
trigonometric reduction formulas:
\begin{itemize}
\item (ST1)
$$\int \tan^n(u)\ du = \frac{\tan^{n-1}(u)}{n-1} - \int \tan^{n-2}(u)\ du.$$
\item (ST2)
$$\int \tan(u)\ du = -\ln|\cos(u)| + C$$
\item (ST3)
$$\int \sec^n(u)\ du = \frac{\tan(u)\sec^{n-2}(u)}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2}(u)\ du.$$
\item (ST4)
$$\int \sec(u)\ du = \ln|\sec(u) + \tan(u)| + C$$
\item For integrals involving both secants and tangents, note that the indentity
$1 + \tan^2(x) = \sec^2(x)$ shows that even powers of secants can be converted
to even powers of tangents and vice versa.
Many integrals involving $\sec(u)$ and $\tan(u)$ together can also be handled by 
rewriting $\tan(u) = \frac{\sin(u)}{\cos(u)}$, $\sec(u) = \frac{1}{\cos(u)}$ and applying
the $\sin(u),\cos(u)$ reductions from last time.  
\end{itemize}
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\noindent
{\it Questions}
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For each of the following integrals, decide which of the reduction formulas above applies,
determine the appropriate $n$ (and $m$ in the last two).  Then apply the formula and
complete the computation.
\begin{enumerate}
\item $\displaystyle \int \sec^3(5x)\ dx$
\item $\displaystyle \int \tan^6(3x)\ dx$
\item $\displaystyle \int \tan^2(4x) \sec^3(4x)\ dx$ (Use $\tan^2(u) = 1 + \sec^2(u)$, then ST3.)
\item $\displaystyle \int \tan^3(5x)\sec^5(5x)\ dx$ (Convert to powers of $\sin(u)$ and $\cos(u)$.) 

\end{enumerate}
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\caption{The region for problem 4.}
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