\def\wsn{32}
\input worksheet.tex

\bigskip
\item{1.}  Prove that the solutions to the equation $ax^2+bx+c=0$ are
$$
x={-b\pm\sqrt{b^2-4ac}\over2a}{\atop.}
$$
Hint: Complete the square.

\bigskip\bigskip\bigskip
\item{2.}  Suppose $h$, $k$, and $r$ are constants with $k/r>0$.  

\medskip
\itemitem{a)}  Find the antidervative $\displaystyle{\int
{1\over r(x+h)^2+k}\,dx}$ using the substitution 
$\displaystyle{\sqrt{k\over r}\tan\theta=x+h}$.

\medskip
\itemitem{b)}  Check your result by differentiating.

\medskip
\itemitem{c)}  Suppose $a$, $b$, and $c$ are constants with
$b^2-4ac<0$.  Find $\displaystyle{\int{1\over ax^2+bx+c}\,dx}$.
 
\bigskip\bigskip\bigskip
\item{3.}  Let $s$, $p$, and $q$ be constants with $s\neq0$ and $p\neq q$.

\medskip
\itemitem{a)}  Show that $\displaystyle{{1\over s(x+p)(x+q)}
={1\over s(p-q)}\left[{1\over x+q}-{1\over x+p}\right]{\atop.}}$

\medskip
\itemitem{b)}  Find the antiderivative $\displaystyle{\int{1\over
s(x+p)(x+q)}\,dx.}$

\medskip
\itemitem{c)}  Check your result by differentiating.

\medskip
\itemitem{d)}  Suppose $a$, $b$, and $c$ are constants with
$b^2-4ac>0$.  Find $\displaystyle{\int{1\over ax^2+bx+c}\,dx}$.

\bigskip\bigskip\bigskip
\item{4.}  Suppose $a$, $b$, and $c$ are constants with
$b^2-4ac=0$.  Find $\displaystyle{\int{1\over ax^2+bx+c}\,dx}$.

\bigskip\bigskip\bigskip
\item{5.}  Summarize your results from Problems 2-4.

\bigskip\bigskip\bigskip
\item{6.}  Evaluate the following integrals:
$$
\pt a \int{x^2-1\over x^2+1}\,dx\hskip1in
\pt b \int{x+1\over x+2}\,dx\hskip1in
\pt c \int{x^3+1\over x^2+x+1}\,dx
$$

\bigskip\bigskip\bigskip
\item{7.}  Find all continuous functions $f(x)$ which satisfy the
equation
$$
(f(x))^2=\int_0^xf(t){t\over1+t^2}\,dt.
$$



\bye