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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Practice Day 1 on Trigonometric Integrals}
\centerline{March 12, 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 are the first ``batch of'' these
trigonometric reduction formulas:
\begin{itemize}
\item (SC1)
$$\int \sin^n(u)\ du = \frac{-\sin^{n-1}(u)\cos(u)}{n} + \frac{n-1}{n} \int \sin^{n-2}(u)\ du.$$
\item (SC2)
$$\int \cos^n(u)\ du = \frac{\cos^{n-1}(u)\sin(u)}{n} + \frac{n-1}{n} \int \cos^{n-2}(u)\ du.$$
\item (SC3 and SC4)
\begin{align*}
&\int \sin^n(u)\cos^m(u)\ du \\
&=\begin{cases}
\displaystyle\frac{-\sin^{n-1}(u)\cos^{m+1}(u)}{n + m} + \frac{n-1}{n + m} \int \sin^{n-2}(u)\cos^m(u)\ du&\\
   &\\
\displaystyle \frac{\sin^{n+1}(u)\cos^{m-1}(u)}{n + m} + \frac{m-1}{n + m} \int \sin^n(u)\cos^{m-2}(u)\ du&
\end{cases}
\end{align*}
\end{itemize}
(Note:  SC3 and SC4 give two separate formulas for the same
integral appearing on the first line.  In some cases both alternatives will apply and the choice
of which one to use is up to you.)
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\noindent
{\it Questions}
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For each of the integrals on the back, 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 $\displaystyle \int \cos^3(x)\ dx$
\item $\displaystyle \int \sin^6(x)\ dx$
\item $\displaystyle \int \sin^2(4x) \cos^3(4x)\ dx$ (Let $u = 4x$ and convert to an equivalent integral in terms of $u$.)
\item $\displaystyle \int \frac{\sin^3(5x)}{\cos^4(5x)}\ dx$ (Let $u = 5x$ first; use SC3 once; the remaining integral can
be finished by a different method if you look at it correctly.)

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