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\title{Math 1b. Improper Integrals}

\date{Spring 2006}

\begin{document}

\maketitle





\begin{enumerate}

\item 
Evaluate $\displaystyle \int_1^t e^{-x} \, dx$.  Does this integral make sense for all $t$?  What if $t \to \infty$?

\vfill

\item
Evaluate $\displaystyle \int_1^t \frac{1}{x} \, dx$.  Does this integral make sense for all $t$?  What if $t \to \infty$?

\vfill

\item
Does $\displaystyle \int_1^\infty \frac{1}{x^2} \, dx$ converge?  If so, to what?

\vfill

\item
Find all values of $p$ such that the integral $\displaystyle \int_1^\infty \frac{1}{x^p} \, dx$ converges.

\vfill

\clearpage

\item
Does $\displaystyle \int_{-\infty}^\infty \frac{1}{1 + x^2} \, dx$ converge?  If so, to what?



\vfill

\item
Does $\displaystyle \int_0^1 \frac{x}{\sqrt{1 - x^2}} \, dx$ converge?  If so, to what?



\vfill


\item
Does $\displaystyle \int_1^\infty \frac{1}{x^5 + 1} \, dx$ converge?  Why or why not?

\vfill


\item
Does $\displaystyle \int_0^\infty \frac{1}{e^x + x} \, dx$ converge?  Why or why not?

\vfill

\item
Does $\displaystyle \int_1^\infty \frac{\sin x}{x^2} \, dx$ converge?  Why or why not?

\vfill

\item
Find all values of $p$ such that the integral $\displaystyle \int_1^e \frac{1}{x (\ln )^p} \, dx$ converges.

\vfill


\end{enumerate}



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