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\begin{document}

\pagestyle{empty}

\noindent
\parbox{2in}{\bf Math 1120 Form A}
\hfill {\Large \bf Test \#2} \hfill
\parbox{2in}{\bf \hfill Oct. $5^\mathrm{th}$, 2021}

\vspace{0.15in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}
\hfill {\bf Be sure to show your work!}

\vspace{-0.1in}

\begin{itemize}
\item Double angle: \qquad $\cos^2(x) = \dfrac{1}{2}\left(1+\cos(2x)\right)$, \quad $\sin^2(x) = \dfrac{1}{2}\left(1-\cos(2x)\right)$, \quad $\sin(2x)=2\sin(x)\cos(x)$

\item Angle sum and difference: \quad $\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$, \quad $\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)$,\\ \force \hspace{1.675in} $\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$, \quad $\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)$ 

\item Product to sum: \qquad $2\sin(x)\sin(y)=\cos(x-y)-\cos(x+y)$, \quad $2\cos(x)\cos(y)=\cos(x-y)+\cos(x+y)$,\\ \force \hspace*{1.25in} $2\sin(x)\cos(y) = \sin(x-y)+\sin(x+y)$
\end{itemize}


\noindent {\bf \large 1. (25 points)} Pythagorean Theorem

\begin{enumerate}[(a)]

\item Write down the Pythagorean Theorem trigonometric identity relating sine and cosine. Then write down the other version relating tangent and secant.

\vspace{0.25in}

Identity \#1: \quad \underline{\hspace*{2.25in}} \qquad \qquad
Identity \#2: \quad \underline{\hspace*{2.25in}}

\vspace{0.1in}

\item In each of the following integrals, use the proper (inverse) trigonometric substitution to simplify the integral. {\bf Do NOT} integrate. Just substitute and {\bf simplify}.

\begin{itemize}
\item $\displaystyle \int \dfrac{dx}{x^2\sqrt{x^2+9\strut}}$
%\item $\displaystyle \int \dfrac{dx}{x^2\sqrt{x^2+4\strut}}$

\vspace{1in}

%\item $\displaystyle \int \dfrac{\sqrt{x^2-9\strut}}{x}\,dx$
\item $\displaystyle \int \dfrac{\sqrt{x^2-4\strut}}{x}\,dx$

\vspace{1in}

\item $\displaystyle \int_0^5 \sqrt{25-x^2}\,dx$
%\item $\displaystyle \int_0^{4} \sqrt{16-x^2}\,dx$

\vspace{1in}

\end{itemize}

\item Given $x=3\sin(\theta)$, rewrite $7\theta-4\tan(\theta)$ in terms of $x$.
%\item Given $x=5\sin(\theta)$, rewrite $4\theta-7\tan(\theta)$ in terms of $x$.

\vspace{0.75in}

\end{enumerate}

\noindent {\bf \large 2. (8 points)} Write down the ``forms'' we would use to find the partial fraction decomposition of \quad $\displaystyle \dfrac{3x^5-x^4+9x^2+8}{x(x+6)^3(x^2+x+3)^2}$.
%\noindent {\bf \large 2. (8 points)} Write down the ``forms'' we would use to find the partial fraction decomposition of \quad $\displaystyle \dfrac{3x^5-x^4+9x^2+8}{x^3(x+7)(x^2+x+11)^2}$.

\newpage

\noindent {\bf \large 3. (15 points)} Complete the Square

\begin{enumerate}[(a)]

\item To integrate $\displaystyle \int \dfrac{6x}{\sqrt{1+3x-x^2}}\,dx$ we would need to split the integral into a $u$-substitution integral and an integral where we complete the square in the radical. Split this integral into those two pieces. {\bf Do NOT} integrate.
%\item To integrate $\displaystyle \int \dfrac{10x}{\sqrt{1+4x-x^2}}\,dx$ we would need to split the integral into a $u$-substitution integral and an integral where we complete the square in the radical. Split this integral into those two pieces. {\bf Do NOT} integrate.

\vspace{0.5in}

$\displaystyle \int \dfrac{6x}{\sqrt{1+3x-x^2}}\,dx$ \quad $=$ \quad \underline{\hspace*{2in}} \quad $+$ \quad \underline{\hspace*{2in}}
%$\displaystyle \int \dfrac{10x}{\sqrt{1+4x-x^2}}\,dx$ \quad $=$ \quad \underline{\hspace*{2in}} \quad $+$ \quad \underline{\hspace*{2in}}

\vspace{0.75in}

\item Compute \quad $\displaystyle \int \dfrac{3}{x^2-4x+13}\,dx$
%\item  Compute \quad $\displaystyle \int \dfrac{2}{x^2-6x+13}\,dx$

\vfill

\end{enumerate}

\noindent {\bf \large 4. (18 points)} Integrate!

\begin{enumerate}[(a)]

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

\vfill

%\item $\displaystyle \int_0^{\pi/2} \sin(3x)\sin(2x)\,dx$
%\item $\displaystyle \int_0^{\pi/2} \sin(4x)\sin(3x)\,dx$
%\vfill

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

\vfill

\item $\displaystyle \int_0^1 \mathrm{arctan}(x)\,dx$

\vfill

\end{enumerate}

\newpage

\noindent {\bf \large 5. (34 points)} Integrate!

\begin{enumerate}[(a)]


\item $\displaystyle \int e^x\sin(3x)\,dx$
%\item $\displaystyle \int e^x\cos(3x)\,dx$

\vfill

\item $\displaystyle \int \tan^5(x)\sec^4(x)\,dx$
%\item $\displaystyle \int \tan^3(x)\sec^6(x)\,dx$

\vfill

\item $\displaystyle \int \sqrt{4-x^2}\,dx$
%\item $\displaystyle \int \sqrt{9-x^2}\,dx$

\vfill


\item $\displaystyle \int \dfrac{2x^2+7x+4}{x(x+1)^2}\,dx$
%\item $\displaystyle \int \dfrac{3x^2+1}{x(x-1)^2}\,dx$

\vfill

\end{enumerate}

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\newpage
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\noindent
\parbox{2in}{\bf Math 1120 Form B}
\hfill {\Large \bf Test \#2} \hfill
\parbox{2in}{\bf \hfill Oct. $5^\mathrm{th}$, 2021}

\vspace{0.15in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}
\hfill {\bf Be sure to show your work!}

\vspace{-0.1in}

\begin{itemize}
\item Double angle: \qquad $\cos^2(x) = \dfrac{1}{2}\left(1+\cos(2x)\right)$, \quad $\sin^2(x) = \dfrac{1}{2}\left(1-\cos(2x)\right)$, \quad $\sin(2x)=2\sin(x)\cos(x)$

\item Angle sum and difference: \quad $\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$, \quad $\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)$,\\ \force \hspace{1.675in} $\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$, \quad $\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)$ 

\item Product to sum: \qquad $2\sin(x)\sin(y)=\cos(x-y)-\cos(x+y)$, \quad $2\cos(x)\cos(y)=\cos(x-y)+\cos(x+y)$,\\ \force \hspace*{1.25in} $2\sin(x)\cos(y) = \sin(x-y)+\sin(x+y)$
\end{itemize}


\noindent {\bf \large 1. (25 points)} Pythagorean Theorem

\begin{enumerate}[(a)]

\item Write down the Pythagorean Theorem trigonometric identity relating sine and cosine. Then write down the other version relating tangent and secant.

\vspace{0.25in}

Identity \#1: \quad \underline{\hspace*{2.25in}} \qquad \qquad
Identity \#2: \quad \underline{\hspace*{2.25in}}

\vspace{0.1in}

\item In each of the following integrals, use the proper (inverse) trigonometric substitution to simplify the integral. {\bf Do NOT} integrate. Just substitute and {\bf simplify}.

\begin{itemize}
%\item $\displaystyle \int \dfrac{dx}{x^2\sqrt{x^2+9\strut}}$
\item $\displaystyle \int \dfrac{dx}{x^2\sqrt{x^2+4\strut}}$

\vspace{1in}

\item $\displaystyle \int \dfrac{\sqrt{x^2-9\strut}}{x}\,dx$
%\item $\displaystyle \int \dfrac{\sqrt{x^2-4\strut}}{x}\,dx$

\vspace{1in}

%\item $\displaystyle \int_0^5 \sqrt{25-x^2}\,dx$
\item $\displaystyle \int_0^{4} \sqrt{16-x^2}\,dx$

\vspace{1in}

\end{itemize}

%\item Given $x=3\sin(\theta)$, rewrite $7\theta-4\tan(\theta)$ in terms of $x$.
\item Given $x=5\sin(\theta)$, rewrite $4\theta-7\tan(\theta)$ in terms of $x$.

\vspace{0.75in}

\end{enumerate}

%\noindent {\bf \large 2. (8 points)} Write down the ``forms'' we would use to find the partial fraction decomposition of \quad $\displaystyle \dfrac{3x^5-x^4+9x^2+8}{x(x+6)^3(x^2+x+3)^2}$.
\noindent {\bf \large 2. (8 points)} Write down the ``forms'' we would use to find the partial fraction decomposition of \quad $\displaystyle \dfrac{3x^5-x^4+9x^2+8}{x^3(x+7)(x^2+x+11)^2}$.

\newpage

\noindent {\bf \large 3. (15 points)} Complete the Square

\begin{enumerate}[(a)]

%\item To integrate $\displaystyle \int \dfrac{6x}{\sqrt{1+3x-x^2}}\,dx$ we would need to split the integral into a $u$-substitution integral and an integral where we complete the square in the radical. Split this integral into those two pieces. {\bf Do NOT} integrate.
\item To integrate $\displaystyle \int \dfrac{10x}{\sqrt{1+4x-x^2}}\,dx$ we would need to split the integral into a $u$-substitution integral and an integral where we complete the square in the radical. Split this integral into those two pieces. {\bf Do NOT} integrate.

\vspace{0.5in}

%$\displaystyle \int \dfrac{6x}{\sqrt{1+3x-x^2}}\,dx$ \quad $=$ \quad \underline{\hspace*{2in}} \quad $+$ \quad \underline{\hspace*{2in}}
$\displaystyle \int \dfrac{10x}{\sqrt{1+4x-x^2}}\,dx$ \quad $=$ \quad \underline{\hspace*{2in}} \quad $+$ \quad \underline{\hspace*{2in}}

\vspace{0.75in}

%\item Compute \quad $\displaystyle \int \dfrac{3}{x^2-4x+13}\,dx$
\item  Compute \quad $\displaystyle \int \dfrac{2}{x^2-6x+13}\,dx$

\vfill

\end{enumerate}

\noindent {\bf \large 4. (18 points)} Integrate!

\begin{enumerate}[(a)]

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

\vfill

%\item $\displaystyle \int_0^{\pi/2} \sin(3x)\sin(2x)\,dx$
%\item $\displaystyle \int_0^{\pi/2} \sin(4x)\sin(3x)\,dx$
%\vfill

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

\vfill

\item $\displaystyle \int_0^1 \mathrm{arctan}(x)\,dx$

\vfill

\end{enumerate}

\newpage

\noindent {\bf \large 5. (34 points)} Integrate!

\begin{enumerate}[(a)]


%\item $\displaystyle \int e^x\sin(3x)\,dx$
\item $\displaystyle \int e^x\cos(3x)\,dx$

\vfill

%\item $\displaystyle \int \tan^5(x)\sec^4(x)\,dx$
\item $\displaystyle \int \tan^3(x)\sec^6(x)\,dx$

\vfill

%\item $\displaystyle \int \sqrt{4-x^2}\,dx$
\item $\displaystyle \int \sqrt{9-x^2}\,dx$

\vfill


%\item $\displaystyle \int \dfrac{2x^2+7x+4}{x(x+1)^2}\,dx$
\item $\displaystyle \int \dfrac{3x^2+1}{x(x-1)^2}\,dx$

\vfill


\end{enumerate}



\end{document}