> For the complete documentation index, see [llms.txt](https://overleaf-pro.ayaka.space/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://overleaf-pro.ayaka.space/latex/pl/matematyka/09-integrals-sums-and-limits.md).
# Całki, sumy i granice
## Całki
Wyrażenie całkowe można dodać za pomocą `\int_{lower}^{upper}` polecenie.
Uwaga, że wyrażenie całkowe może wyglądać nieco inaczej w trybie matematyki w tekście i w trybie wyświetlanym.
| Kod LaTeX | Wynik |
| ------------------------------------------- | -------------------------------------------------------------------------------------- |
| `Całka \(\int_{a}^{b} x^2 \,dx\) w tekście` |  |
| `\[ \int_{a}^{b} x^2 \,dx \]` |  |
## Wielokrotne całki
Aby uzyskać całki podwójne/potrójne/wielokrotne oraz całki cykliczne, należy użyć `amsmath` i `esint` (dla całek cyklicznych) pakiety.
| Kod LaTeX | Wynik |
| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- |
| `latex
\begin{gather*}
\iint_V \mu(u,v) \,du\,dv
\\
\iiint_V \mu(u,v,w) \,du\,dv\,dw
\\
\iiiint_V \mu(t,u,v,w) \,dt\,du\,dv\,dw
\\
\idotsint_V \mu(u_1,\dots,u_k) \,du_1 \dots du_k
\end{gather*}
` |  |
| `latex
\[
\oint_V f(s) \,ds
\]
` |  |
## Sumy i iloczyny
Podobnie jak całkę, wyrażenie sumy można dodać za pomocą `\sum_{lower}^{upper}` polecenie.
| Kod LaTeX | Wynik |
| ------------------------------------------------- | ----------------------------------------------------------------------------------------- |
| `Suma $\sum_{n=1}^{\infty} 2^{-n} = 1$ w tekście` |  |
| `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` |  |
W podobny sposób można uzyskać wyrażenie z iloczynem ciągu czynników za pomocą `\prod_{lower}^{upper}` polecenie.
| Kod LaTeX | Wynik |
| ------------------------------------------ | ------------------------------------------------------------------------------------------- |
| `Iloczyn $\prod_{i=a}^{b} f(i)$ w tekście` |  |
| `\[ \prod_{i=a}^{b} f(i) \]` |  |
## Granice
Wyrażenie graniczne można dodać za pomocą `\lim_{lower}` polecenie.
| Kod LaTeX | Wynik |
| -------------------------------------------- | --------------------------------------------------------------------------------- |
| `Granica $\lim_{x\to\infty} f(x)$ w tekście` |  |
| `\[ \lim_{x\to\infty} f(x) \]` |  |
## Przykłady
Oto przykładowy dokument LaTeX:
```latex
\documentclass{article}
\title{Całki, Sumy i Granice}
\author{Overleaf}
\date{}
\usepackage{amsmath}
\begin{document}
\maketitle
\section{Całki}
Całka \(\int_{a}^{b} x^2 dx\) w tekście.
\medskip
Ta sama całka w trybie wyświetlanym:
\[
\int_{a}^{b} x^2 \,dx
\]
oraz wielokrotne całki:
\begin{gather*}
\iint_V \mu(u,v) \,du\,dv
\\
\iiint_V \mu(u,v,w) \,du\,dv\,dw
\\
\iiiint_V \mu(t,u,v,w) \,dt\,du\,dv\,dw
\\
\idotsint_V \mu(u_1,\dots,u_k) \,du_1 \dots du_k
\\
\oint_V f(s) \,ds
\end{gather*}
\section{Sumy i iloczyny}
Suma \(\sum_{n=1}^{\infty} 2^{-n} = 1\) w tekście.
Ta sama suma w trybie wyświetlanym:
\[
\sum_{n=1}^{\infty} 2^{-n} = 1
\]
Iloczyn \(\prod_{i=a}^{b} f(i)\) w tekście.
Ten sam iloczyn w trybie wyświetlanym:
\[
\prod_{i=a}^{b} f(i)
\]
\section{Granice}
Granica \(\lim_{x\to\infty} f(x)\) w tekście.
Ta sama granica w trybie wyświetlanym:
\[
\lim_{x\to\infty} f(x)
\]
\end{document}
```
[Otwórz ten przykład w Overleaf](https://www.overleaf.com/docs?engine=lualatex\&snip_name=Integrals%2C+Sums+and+Limits\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Ctitle%7BIntegrals%2C+Sums+and+Limits%7D%0A%5Cauthor%7BOverleaf%7D%0A%5Cdate%7B%7D%0A%5Cusepackage%7Bamsmath%7D%0A%0A%5Cbegin%7Bdocument%7D%0A%0A%5Cmaketitle%0A%0A%5Csection%7BIntegrals%7D%0A%0AIntegral+%5C%28%5Cint_%7Ba%7D%5E%7Bb%7D+x%5E2+dx%5C%29+inside+text.%0A%0A%5Cmedskip%0A%0AThe+same+integral+on+display%3A%0A%5C%5B%0A++++%5Cint_%7Ba%7D%5E%7Bb%7D+x%5E2+%5C%2Cdx%0A%5C%5D%0Aand+multiple+integrals%3A%0A%5Cbegin%7Bgather%2A%7D%0A++++%5Ciint_V+%5Cmu%28u%2Cv%29+%5C%2Cdu%5C%2Cdv%0A%5C%5C%0A++++%5Ciiint_V+%5Cmu%28u%2Cv%2Cw%29+%5C%2Cdu%5C%2Cdv%5C%2Cdw%0A%5C%5C%0A++++%5Ciiiint_V+%5Cmu%28t%2Cu%2Cv%2Cw%29+%5C%2Cdt%5C%2Cdu%5C%2Cdv%5C%2Cdw%0A%5C%5C%0A++++%5Cidotsint_V+%5Cmu%28u_1%2C%5Cdots%2Cu_k%29+%5C%2Cdu_1+%5Cdots+du_k%0A%5C%5C%0A++++%5Coint_V+f%28s%29+%5C%2Cds%0A%5Cend%7Bgather%2A%7D%0A%0A%5Csection%7BSums+and+products%7D%0A%0ASum+%5C%28%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+2%5E%7B-n%7D+%3D+1%5C%29+inside+text.%0A%0AThe+same+sum+on+display%3A%0A%5C%5B%0A++++%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+2%5E%7B-n%7D+%3D+1%0A%5C%5D%0A%0AProduct+%5C%28%5Cprod_%7Bi%3Da%7D%5E%7Bb%7D+f%28i%29%5C%29+inside+text.%0A%0AThe+same+product+on+display%3A%0A%5C%5B%0A++++%5Cprod_%7Bi%3Da%7D%5E%7Bb%7D+f%28i%29%0A%5C%5D%0A%0A%5Csection%7BLimits%7D%0A%0ALimit+%5C%28%5Clim_%7Bx%5Cto%5Cinfty%7D+f%28x%29%5C%29+inside+text.%0A%0AThe+same+limit+on+display%3A%0A%5C%5B%0A++++%5Clim_%7Bx%5Cto%5Cinfty%7D+f%28x%29%0A%5C%5D%0A%0A%5Cend%7Bdocument%7D)
## Dalsza lektura
Więcej informacji znajdziesz w
* [Wyrażenia matematyczne](/latex/pl/matematyka/01-mathematical-expressions.md)
* [Indeksy dolne i górne](/latex/pl/matematyka/02-subscripts-and-superscripts.md)
* [Ułamki i dwumiany](/latex/pl/matematyka/05-fractions-and-binomials.md)
* [Styl wyświetlania w trybie matematycznym](/latex/pl/matematyka/10-display-style-in-math-mode.md)
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