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\begin{center}
\textbf{\Large Math 1234} 
\end{center}

\begin{enumerate}

\item Evaluate the following integrals:
  \begin{enumerate}
  \item $\displaystyle \int (3x-1)^5 \, dx$
  \item $\displaystyle \int (\sqrt{3t+1} + \sin t) \, dt$
  \item $\displaystyle \int (4x^2-12x+9)^{2/3} \, dx $
  \end{enumerate}

\item Given the following system of inequalities:
\begin{align*}
   -x+ y  &\geq 1\\ 
    3x+2y &\geq 2\\
     y    &\leq 5   
\end{align*}

\begin{enumerate}
  \item Graph the system.
  \item Is the feasible region of the system bounded or unbounded?
  \end{enumerate}

\item Let $A, B$, and $C$ be sets.  
Show that $A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)$.

{\bf Solution.} Let $x \in A \cap (B\cup C)$. Now, 
\begin{align*}
  x \in A \cap (B \cup C) 
      & \Rightarrow x \in A \text{ and } x \in (B \cup C)\\
      & \Rightarrow x \in A \text{ and } (x \in B \text{ or } x \in C)\\
      & \Rightarrow (x \in A \text{ and } x \in B) \text{ or } (x \in A \text{ and } x \in C)\\
      & \Rightarrow x \in (A \cap B) \cup (A \cap C).
\end{align*}

\end{enumerate}
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