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\centerline{MATH 136 -- Calculus 2}
\centerline{First Practice on Improper Integrals}
\centerline{February 25, 2020}
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\noindent
{\it Background}
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Whenever $a = -\infty$ or $b = +\infty$ or both, \emph{or} the function 
$f(x)$ has a discontinuity in the interval $[a,b]$, the integral 
${\displaystyle \int_a^b f(x)\ dx}$
is said to be an \emph{improper integral}. Improper integrals are always
 handled by \emph{taking limits of ``ordinary'' integrals}.  
\begin{itemize}
\item We say $\int_a^\infty f(x)\ dx$ \emph{converges}
if the limit 
$$\lim_{b \to \infty} \int_a^b f(x)\ dx$$ 
exists and we say the integral \emph{diverges} otherwise.
\item We say $\int_{-\infty}^b f(x)\ dx$ \emph{converges}
if the limit 
$$\lim_{a \to -\infty} \int_a^b f(x)\ dx$$ 
exists and we say the integral \emph{diverges} otherwise.
\item We say $\int_{-\infty}^\infty f(x)\ dx$ \emph{converges}
if, splitting at any finite value $c$ (often at $c = 0$), \emph{both of the limits} here: 
$$\lim_{a \to -\infty} \int_a^c f(x)\ dx + \lim_{b \to +\infty} \int_c^b f(x)\ dx$$ 
exist and we say the integral \emph{diverges} otherwise.  (Note this says that if either one
of the two limits does not exist, then the integral diverges.)
\item If $f$ is discontinuous at $a$, then we say $\int_a^b f(x)\ dx$ \emph{converges}
if the limit 
$$\lim_{c \to a^+} \int_c^b f(x)\ dx$$ 
exists and we say the integral \emph{diverges} otherwise.
\item If $f$ is discontinuous at $b$, then we say $\int_a^b f(x)\ dx$ \emph{converges}
if the limit 
$$\lim_{c \to b^-} \int_a^c f(x)\ dx$$ 
exists and we say the integral \emph{diverges} otherwise.
\item If $f$ is discontinuous at $d$ in $(a,b)$, then we say $\int_a^b f(x)\ dx$ \emph{converges}
if \emph{both of the limits} here: 
$$\lim_{c \to d^-} \int_a^c f(x)\ dx + \lim_{c \to d^+} \int_c^b f(x)\ dx$$ 
exist and we say the integral \emph{diverges} otherwise.  (Note this says that if either one
of the two limits does not exist, then the integral diverges.)
\end{itemize}
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\noindent
{\it Questions}
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For each of the following,
\begin{enumerate}
\item[(i)] Explain why the integral is improper.
\item[(ii)] Set up the appropriate limit(s) and evaluate.
\item[(iii)]  Say whether the integral converges or diverges.
\end{enumerate}
\begin{enumerate}
\item $\displaystyle \int_{-\infty}^2 e^{10x}\ dx$
\item $\displaystyle \int_4^{\infty} \frac{1}{x^2 + 16}\ dx$
\item $\displaystyle \int_{-\infty}^\infty \frac{1}{x^2 + 16}\ dx$
\item $\displaystyle \int_1^2 \frac{1}{\sqrt[3]{x - 1}}\ dx$
\item $\displaystyle \int_0^2 \frac{1}{x^2 - 5x + 6}\ dx$
\item $\displaystyle \int_1^3 \frac{1}{x^2 - 4x + 4}\ dx$
\end{enumerate}
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