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\noindent {\bf Question}

\noindent Determine whether the improper integral
\[ \int_0^1 \frac{e^{\sqrt{x}}}{\sqrt{x}} {\rm d}x \]
converges or diverges.  In the case that it converges, determine
its value.

\medskip

\noindent {\bf Answer}

\noindent \begin{eqnarray*} \int_0^1 \frac{e^{\sqrt{x}}}{\sqrt{x}}
{\rm d}x & = & \lim_{\varepsilon\rightarrow 0+} \int_\varepsilon^1
\frac{e^{\sqrt{x}}}{\sqrt{x}} {\rm d}x \\ & = &
\lim_{\varepsilon\rightarrow 0+} 2e^{\sqrt{x}}
\left|_\varepsilon^1 \right. \\ & = & \lim_{\varepsilon\rightarrow
0+} ( 2e -2^{\sqrt{\varepsilon}} ) = 2e -2,
\end{eqnarray*}
which converges.


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