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\begin_layout Standard
Dobovišek.
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\begin_layout Enumerate
Poišči definicijsko območje funkcije,
 podane z integralom
\begin_inset Formula 
\[
f\left(y\right)=\int_{0}^{1}\frac{dx}{x^{2}+y^{2}}.
\]

\end_inset


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\begin_deeper
\begin_layout Standard
Rešitev:
 
\begin_inset Formula $D_{f}=\mathbb{R}\setminus\left\{ 0\right\} $
\end_inset

.
\end_layout

\begin_layout Standard
Razlaga:
 Za 
\begin_inset Formula $y=0$
\end_inset

 je integral 
\begin_inset Formula $\int_{0}^{1}\frac{dx}{x^{2}}$
\end_inset

.
 Nedoločeni je 
\begin_inset Formula $\int\frac{dx}{x^{2}}dx=\frac{-1}{x}$
\end_inset

,
 torej 
\begin_inset Formula $\int_{0}^{1}\frac{dx}{x^{2}}=\left.\frac{-1}{x}\right|_{0}^{1}=-1+\lim_{x\to0}\frac{1}{x}\not\in\mathbb{R}$
\end_inset

.
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\begin_layout Enumerate
Izračunaj limiti:
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\begin_inset ERT
status open

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\backslash
begin{multicols}{2}
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\end_inset


\end_layout

\begin_layout Enumerate
\begin_inset Formula 
\[
\lim_{R\to\infty}\int_{0}^{\pi/2}e^{-R\sin x}dx,
\]

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Rešitev:
\begin_inset Formula 
\[
\int_{0}^{\pi/2}
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula 
\[
\lim_{\alpha\to0}\int_{\alpha}^{1+\alpha}\frac{dx}{1+x^{2}+\alpha^{2}}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

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end{multicols}
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