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%%%%%%%%%%%% New theorems %%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{theorem}{Î˜ÎµÏŽÏÎ·Î¼Î±}[section]
\newtheorem{lemma}[theorem]{Î›Î®Î¼Î¼Î±}
\newtheorem{proposition}[theorem]{Î&nbsp;ÏÏŒÏ„Î±ÏƒÎ·}
\newtheorem{application}[theorem]{Î•Ï†Î±ÏÎ¼Î¿Î³Î®}
\newtheorem{corollary}[theorem]{Î&nbsp;ÏŒÏÎ¹ÏƒÎ¼Î±}
\newtheorem{definition}[theorem]{ÎŸÏÎ¹ÏƒÎ¼ÏŒÏ‚}
\newtheorem{exercise}[theorem]{Î†ÏƒÎºÎ·ÏƒÎ·}
\newtheorem{example}[theorem]{Î&nbsp;Î±ÏÎ¬Î´ÎµÎ¹Î³Î¼Î±}
\newtheorem{examples}[theorem]{Î&nbsp;Î±ÏÎ±Î´ÎµÎ¯Î³Î¼Î±Ï„Î±}
\newtheorem{problem}[theorem]{Î&nbsp;ÏÏŒÎ²Î»Î·Î¼Î±}
\newtheorem{remark}[theorem]{Î&nbsp;Î±ÏÎ±Ï„Î®ÏÎ·ÏƒÎ·}
\newtheorem{remarks}[theorem]{Î&nbsp;Î±ÏÎ±Ï„Î·ÏÎ®ÏƒÎµÎ¹Ï‚}
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\begin{document}
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\textbf{Î‘Ï€ÎµÎ¹ÏÎ¿ÏƒÏ„Î¹ÎºÏŒÏ‚ Î›Î¿Î³Î¹ÏƒÎ¼ÏŒÏ‚ Î™Î™}
\textbf{Î&nbsp;ÏÏŒÏ‡ÎµÎ¹ÏÎµÏ‚ Î£Î·Î¼ÎµÎ¹ÏŽÏƒÎµÎ¹Ï‚}
\textbf{Î¤Î¼Î®Î¼Î± ÎœÎ±Î¸Î·Î¼Î±Ï„Î¹ÎºÏŽÎ½}
\textbf{Î&nbsp;Î±Î½ÎµÏ€Î¹ÏƒÏ„Î®Î¼Î¹Î¿ Î‘Î¸Î·Î½ÏŽÎ½}
\textbf{2010--11}

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\chapter{Î¥Ï€Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯ÎµÏ‚ ÎºÎ±Î¹ Î²Î±ÏƒÎ¹ÎºÎ­Ï‚ Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯ÎµÏ‚}
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\section{Î¥Ï€Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯ÎµÏ‚}

\begin{definition} \upshape ÎˆÏƒÏ„Ï‰ $(a_n)$ Î¼Î¹Î± Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯Î± Ï€ÏÎ±Î³Î¼Î±Ï„Î¹ÎºÏŽÎ½ Î±ÏÎ¹Î¸Î¼ÏŽÎ½.
Î— Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯Î± $(b_n)$ Î»Î­Î³ÎµÏ„Î±Î¹ \textit{Ï…Ï€Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯Î±} Ï„Î·Ï‚ $(a_n)$ Î±Î½ Ï…Ï€Î¬ÏÏ‡ÎµÎ¹
Î³Î½Î·ÏƒÎ¯Ï‰Ï‚ Î±ÏÎ¾Î¿Ï…ÏƒÎ± Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯Î± Ï†Ï…ÏƒÎ¹ÎºÏŽÎ½ Î±ÏÎ¹Î¸Î¼ÏŽÎ½ $k_1 &lt; k_2&lt; \cdots &lt; k_n &lt;
k_{n+1}&lt;\cdots $ ÏŽÏƒÏ„Îµ
$$b_n = a_{k_n}\;\hbox{ Î³Î¹Î± ÎºÎ¬Î¸Îµ }\;n \in {\mathbb N}.\leqno (1.1.1)$$
ÎœÎµ Î¬Î»Î»Î± Î»ÏŒÎ³Î¹Î±, Î¿Î¹ ÏŒÏÎ¿Î¹ Ï„Î·Ï‚ $(b_n)$ ÎµÎ¯Î½Î±Î¹ Î¿Î¹ $a_{k_1}, a_{k_2},
\ldots, a_{k_n}, \ldots $, ÏŒÏ€Î¿Ï… $k_1 &lt; k_2&lt; \cdots &lt; k_n &lt;
k_{n+1}&lt;\cdots$. Î“ÎµÎ½Î¹ÎºÎ¬, Î¼Î¹Î± Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯Î± Î­Ï‡ÎµÎ¹ Ï€Î¿Î»Î»Î­Ï‚ (ÏƒÏ…Î½Î®Î¸Ï‰Ï‚ Î¬Ï€ÎµÎ¹ÏÎµÏ‚
Ï„Î¿ Ï€Î»Î®Î¸Î¿Ï‚) Î´Î¹Î±Ï†Î¿ÏÎµÏ„Î¹ÎºÎ­Ï‚ Ï…Ï€Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯ÎµÏ‚.
\end{definition}


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\chapter{Î£ÎµÎ¹ÏÎ­Ï‚ Ï€ÏÎ±Î³Î¼Î±Ï„Î¹ÎºÏŽÎ½ Î±ÏÎ¹Î¸Î¼ÏŽÎ½}
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\section{Î£ÏÎ³ÎºÎ»Î¹ÏƒÎ· ÏƒÎµÎ¹ÏÎ¬Ï‚}

\begin{definition} \upshape ÎˆÏƒÏ„Ï‰ $(a_k)$ Î¼Î¹Î± Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯Î± Ï€ÏÎ±Î³Î¼Î±Ï„Î¹ÎºÏŽÎ½
Î±ÏÎ¹Î¸Î¼ÏŽÎ½. Î˜ÎµÏ‰ÏÎ¿ÏÎ¼Îµ Ï„Î·Î½ Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯Î± $$s_n=a_1+\cdots +a_n.\leqno
(2.1.1)$$ Î”Î·Î»Î±Î´Î®,
$$s_1=a_1,\  s_2=a_1+a_2,\  s_3=a_1+a_2+a_3,\  \ldots \leqno (2.1.2)$$
Î¤Î¿ ÏƒÏÎ¼Î²Î¿Î»Î¿ $\sum_{k=1}^{\infty }a_k$ ÎµÎ¯Î½Î±Î¹ Î· \textit{ÏƒÎµÎ¹ÏÎ¬} Î¼Îµ
$k$-Î¿ÏƒÏ„ÏŒ ÏŒÏÎ¿ Ï„Î¿Î½ $a_k$. Î¤Î¿ Î¬Î¸ÏÎ¿Î¹ÏƒÎ¼Î± $s_n=\sum_{k=1}^na_k$
ÎµÎ¯Î½Î±Î¹ Ï„Î¿ \textit{$n$-Î¿ÏƒÏ„ÏŒ Î¼ÎµÏÎ¹ÎºÏŒ Î¬Î¸ÏÎ¿Î¹ÏƒÎ¼Î±} Ï„Î·Ï‚ ÏƒÎµÎ¹ÏÎ¬Ï‚
$\sum_{k=1}^{\infty }a_k$ ÎºÎ±Î¹ Î· $(s_n)$ ÎµÎ¯Î½Î±Î¹ Î· {\it
Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯Î± Ï„Ï‰Î½ Î¼ÎµÏÎ¹ÎºÏŽÎ½ Î±Î¸ÏÎ¿Î¹ÏƒÎ¼Î¬Ï„Ï‰Î½} Ï„Î·Ï‚ ÏƒÎµÎ¹ÏÎ¬Ï‚ $ \sum_{k =
1}^{\infty }a_k$.

Î‘Î½ Î· $(s_n)$ ÏƒÏ…Î³ÎºÎ»Î¯Î½ÎµÎ¹ ÏƒÎµ ÎºÎ¬Ï€Î¿Î¹Î¿Î½ Ï€ÏÎ±Î³Î¼Î±Ï„Î¹ÎºÏŒ Î±ÏÎ¹Î¸Î¼ÏŒ $s$, Ï„ÏŒÏ„Îµ
Î³ÏÎ¬Ï†Î¿Ï…Î¼Îµ
$$s = a_1 + a_2 + \cdots + a_n + \cdots\  \hbox{Î®}\  s=\sum_{k=1}^{\infty }a_k\leqno (2.1.3)$$
ÎºÎ±Î¹ Î»Î­Î¼Îµ ÏŒÏ„Î¹ Î· ÏƒÎµÎ¹ÏÎ¬ \textit{ÏƒÏ…Î³ÎºÎ»Î¯Î½ÎµÎ¹} (ÏƒÏ„Î¿ $s$), Ï„Î¿ Î´Îµ ÏŒÏÎ¹Î¿
$s=\lim_{n\to\infty }s_n$ ÎµÎ¯Î½Î±Î¹ Ï„Î¿ \textit{Î¬Î¸ÏÎ¿Î¹ÏƒÎ¼Î±} Ï„Î·Ï‚ ÏƒÎµÎ¹ÏÎ¬Ï‚.
\end{definition}


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\chapter{ÎŸÎ»Î¿ÎºÎ»Î®ÏÏ‰Î¼Î± \textlatin{Riemann}}
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\section{ÎŸ Î¿ÏÎ¹ÏƒÎ¼ÏŒÏ‚ Ï„Î¿Ï… \textlatin{Darboux}}

Î£Îµ Î±Ï…Ï„Î®Î½ Ï„Î·Î½ Ï€Î±ÏÎ¬Î³ÏÎ±Ï†Î¿ Î´Î¯Î½Î¿Ï…Î¼Îµ Ï„Î¿Î½ Î¿ÏÎ¹ÏƒÎ¼ÏŒ Ï„Î¿Ï… Î¿Î»Î¿ÎºÎ»Î·ÏÏŽÎ¼Î±Ï„Î¿Ï‚
\textlatin{Riemann} Î³Î¹Î± \textbf{Ï†ÏÎ±Î³Î¼Î­Î½ÎµÏ‚} ÏƒÏ…Î½Î±ÏÏ„Î®ÏƒÎµÎ¹Ï‚ Ï€Î¿Ï… Î¿ÏÎ¯Î¶Î¿Î½Ï„Î±Î¹ ÏƒÎµ Î­Î½Î±
ÎºÎ»ÎµÎ¹ÏƒÏ„ÏŒ Î´Î¹Î¬ÏƒÏ„Î·Î¼Î±. Î“Î¹Î± Î¼Î¹Î± Ï†ÏÎ±Î³Î¼Î­Î½Î· ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ· $f:[a,b]\to {\mathbb
R}$ Î¼Îµ Î¼Î· Î±ÏÎ½Î·Ï„Î¹ÎºÎ­Ï‚ Ï„Î¹Î¼Î­Ï‚, Î¸Î± Î¸Î­Î»Î±Î¼Îµ Ï„Î¿ Î¿Î»Î¿ÎºÎ»Î®ÏÏ‰Î¼Î± Î½Î± Î´Î¯Î½ÎµÎ¹ Ï„Î¿
ÎµÎ¼Î²Î±Î´ÏŒÎ½ Ï„Î¿Ï… Ï‡Ï‰ÏÎ¯Î¿Ï… Ï€Î¿Ï… Ï€ÎµÏÎ¹ÎºÎ»ÎµÎ¯ÎµÏ„Î±Î¹ Î±Î½Î¬Î¼ÎµÏƒÎ± ÏƒÏ„Î¿ Î³ÏÎ¬Ï†Î·Î¼Î± Ï„Î·Ï‚
ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ·Ï‚, Ï„Î¿Î½ Î¿ÏÎ¹Î¶ÏŒÎ½Ï„Î¹Î¿ Î¬Î¾Î¿Î½Î± $y=0$ ÎºÎ±Î¹ Ï„Î¹Ï‚ ÎºÎ±Ï„Î±ÎºÏŒÏÏ…Ï†ÎµÏ‚ ÎµÏ…Î¸ÎµÎ¯ÎµÏ‚
$x=a$ ÎºÎ±Î¹ $x=b$.

\begin{definition} \upshape (Î±) ÎˆÏƒÏ„Ï‰ $[a,b]$ Î­Î½Î± ÎºÎ»ÎµÎ¹ÏƒÏ„ÏŒ Î´Î¹Î¬ÏƒÏ„Î·Î¼Î±.
\textbf{Î”Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ·} Ï„Î¿Ï… $[a,b]$ Î¸Î± Î»Î­Î¼Îµ ÎºÎ¬Î¸Îµ Ï€ÎµÏ€ÎµÏÎ±ÏƒÎ¼Î­Î½Î¿ Ï…Ï€Î¿ÏƒÏÎ½Î¿Î»Î¿
$$P
=\{ x_0,x_1,\ldots ,x_n\}\leqno (4.1.1)$$ Ï„Î¿Ï… $[a,b]$ Î¼Îµ $x_0=a$
ÎºÎ±Î¹ $x_n=b$. Î˜Î± Ï…Ï€Î¿Î¸Î­Ï„Î¿Ï…Î¼Îµ Ï€Î¬Î½Ï„Î± ÏŒÏ„Î¹ Ï„Î± $x_k\in P $ ÎµÎ¯Î½Î±Î¹
Î´Î¹Î±Ï„ÎµÏ„Î±Î³Î¼Î­Î½Î± Ï‰Ï‚ ÎµÎ¾Î®Ï‚:
$$a=x_0&lt;x_1&lt;\cdots &lt;x_k&lt;x_{k+1}&lt;\cdots &lt;x_n=b.\leqno (4.1.2)$$
Î˜Î± Î³ÏÎ¬Ï†Î¿Ï…Î¼Îµ
$$P =\{ a=x_0&lt;x_1&lt;\cdots &lt;x_n=b\}\leqno (4.1.3)$$ Î³Î¹Î± Î½Î± Ï„Î¿Î½Î¯ÏƒÎ¿Ï…Î¼Îµ Î±Ï…Ï„Î®Î½
Î±ÎºÏÎ¹Î²ÏŽÏ‚ Ï„Î· Î´Î¹Î¬Ï„Î±Î¾Î·. Î&nbsp;Î±ÏÎ±Ï„Î·ÏÎ®ÏƒÏ„Îµ ÏŒÏ„Î¹ Î±Ï€ÏŒ Ï„Î¿Î½ Î¿ÏÎ¹ÏƒÎ¼ÏŒ, ÎºÎ¬Î¸Îµ Î´Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ·
$ P $ Ï„Î¿Ï… $[a,b]$ Ï€ÎµÏÎ¹Î­Ï‡ÎµÎ¹ Ï„Î¿Ï…Î»Î¬Ï‡Î¹ÏƒÏ„Î¿Î½ Î´ÏÎ¿ ÏƒÎ·Î¼ÎµÎ¯Î±: Ï„Î¿ $a$ ÎºÎ±Î¹ Ï„Î¿
$b$ (Ï„Î± Î¬ÎºÏÎ± Ï„Î¿Ï… $[a,b]$).



 (Î²) ÎšÎ¬Î¸Îµ Î´Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ· $ P =\{ a=x_0&lt;x_1&lt;\cdots &lt;x_n=b\}$
Ï‡Ï‰ÏÎ¯Î¶ÎµÎ¹ Ï„Î¿ $[a,b]$ ÏƒÎµ $n$ Ï…Ï€Î¿Î´Î¹Î±ÏƒÏ„Î®Î¼Î±Ï„Î± $[x_k,x_{k+1}]$,
$k=0,1,\ldots ,n-1$. ÎŸÎ½Î¿Î¼Î¬Î¶Î¿Ï…Î¼Îµ \textbf{Ï€Î»Î¬Ï„Î¿Ï‚} Ï„Î·Ï‚ Î´Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ·Ï‚ $ P $
Ï„Î¿ Î¼ÎµÎ³Î±Î»ÏÏ„ÎµÏÎ¿ Î±Ï€ÏŒ Ï„Î± Î¼Î®ÎºÎ· Î±Ï…Ï„ÏŽÎ½ Ï„Ï‰Î½ Ï…Ï€Î¿Î´Î¹Î±ÏƒÏ„Î·Î¼Î¬Ï„Ï‰Î½. Î”Î·Î»Î±Î´Î®, Ï„Î¿
Ï€Î»Î¬Ï„Î¿Ï‚ Ï„Î·Ï‚ Î´Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ·Ï‚ Î¹ÏƒÎ¿ÏÏ„Î±Î¹ Î¼Îµ
$$\| P\|:=\max\{ x_1-x_0,x_2-x_1,\ldots ,x_n-x_{n-1}\}.\leqno (4.1.4)$$
Î&nbsp;Î±ÏÎ±Ï„Î·ÏÎ®ÏƒÏ„Îµ ÏŒÏ„Î¹ Î´ÎµÎ½ Î±Ï€Î±Î¹Ï„Î¿ÏÎ¼Îµ Î½Î± Î¹ÏƒÎ±Ï€Î­Ï‡Î¿Ï…Î½ Ï„Î± $x_k$ (Ï„Î± $n$
Ï…Ï€Î¿Î´Î¹Î±ÏƒÏ„Î®Î¼Î±Ï„Î± Î´ÎµÎ½ Î­Ï‡Î¿Ï…Î½ Î±Ï€Î±ÏÎ±Î¯Ï„Î·Ï„Î± Ï„Î¿ Î¯Î´Î¹Î¿ Î¼Î®ÎºÎ¿Ï‚).



 (Î³) Î— Î´Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ· $ P_1$ Î»Î­Î³ÎµÏ„Î±Î¹ \textbf{ÎµÎºÎ»Î­Ï€Ï„Ï…Î½ÏƒÎ·} Ï„Î·Ï‚ $ P
$ Î±Î½ $ P \subseteq P_1$, Î´Î·Î»Î±Î´Î® Î±Î½ Î· $P_1$ Ï€ÏÎ¿ÎºÏÏ€Ï„ÎµÎ¹ Î±Ï€ÏŒ Ï„Î·Î½ $ P $
Î¼Îµ Ï„Î·Î½ Ï€ÏÎ¿ÏƒÎ¸Î®ÎºÎ· ÎºÎ¬Ï€Î¿Î¹Ï‰Î½ (Ï€ÎµÏ€ÎµÏÎ±ÏƒÎ¼Î­Î½Ï‰Î½ Ï„Î¿ Ï€Î»Î®Î¸Î¿Ï‚) ÏƒÎ·Î¼ÎµÎ¯Ï‰Î½. Î£Îµ Î±Ï…Ï„Î®Î½
Ï„Î·Î½ Ï€ÎµÏÎ¯Ï€Ï„Ï‰ÏƒÎ· Î»Î­Î¼Îµ ÎµÏ€Î¯ÏƒÎ·Ï‚ ÏŒÏ„Î¹ Î· $ P_1$ ÎµÎ¯Î½Î±Î¹ \textit{Î»ÎµÏ€Ï„ÏŒÏ„ÎµÏÎ·} Î±Ï€ÏŒ
Ï„Î·Î½ $ P $.



 (Î´) ÎˆÏƒÏ„Ï‰ $ P_1, P_2$ Î´ÏÎ¿ Î´Î¹Î±Î¼ÎµÏÎ¯ÏƒÎµÎ¹Ï‚ Ï„Î¿Ï… $[a,b]$. Î— 
\textbf{ÎºÎ¿Î¹Î½Î® ÎµÎºÎ»Î­Ï€Ï„Ï…Î½ÏƒÎ·} Ï„Ï‰Î½ $ P_1, P_2$ ÎµÎ¯Î½Î±Î¹ Î· Î´Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ· $ P = P_1\cup
P_2$. Î•ÏÎºÎ¿Î»Î± Î²Î»Î­Ï€Î¿Ï…Î¼Îµ ÏŒÏ„Î¹ Î· $ P $ ÎµÎ¯Î½Î±Î¹ Î´Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ· Ï„Î¿Ï… $[a,b]$ ÎºÎ±Î¹
ÏŒÏ„Î¹ Î±Î½ $ P^{\prime }$ ÎµÎ¯Î½Î±Î¹ Î¼Î¹Î± Î´Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ· Î»ÎµÏ€Ï„ÏŒÏ„ÎµÏÎ· Ï„ÏŒÏƒÎ¿ Î±Ï€ÏŒ Ï„Î·Î½ $
P_1$ ÏŒÏƒÎ¿ ÎºÎ±Î¹ Î±Ï€ÏŒ Ï„Î·Î½ $ P_2$ Ï„ÏŒÏ„Îµ $ P^{\prime }\supseteq P $
(Î´Î·Î»Î±Î´Î®, Î· $ P = P_1\cup P_2$ ÎµÎ¯Î½Î±Î¹ Î· Î¼Î¹ÎºÏÏŒÏ„ÎµÏÎ· Î´Ï…Î½Î±Ï„Î® Î´Î¹Î±Î¼Î­ÏÎ¹ÏƒÎ·
Ï„Î¿Ï… $[a,b]$ Ï€Î¿Ï… ÎµÎºÎ»ÎµÏ€Ï„ÏÎ½ÎµÎ¹ Ï„Î±Ï…Ï„ÏŒÏ‡ÏÎ¿Î½Î± Ï„Î·Î½ $ P_1$ ÎºÎ±Î¹ Ï„Î·Î½ $ P_2$).
\end{definition}


\section{Î™Î´Î¹ÏŒÏ„Î·Ï„ÎµÏ‚ Ï„Î¿Ï… Î¿Î»Î¿ÎºÎ»Î·ÏÏŽÎ¼Î±Ï„Î¿Ï‚ \textlatin{Riemann}}

Î£Îµ Î±Ï…Ï„Î® Ï„Î·Î½ Ï€Î±ÏÎ¬Î³ÏÎ±Ï†Î¿ Î±Ï€Î¿Î´ÎµÎ¹ÎºÎ½ÏÎ¿Ï…Î¼Îµ Î±Ï…ÏƒÏ„Î·ÏÎ¬ Î¼ÎµÏÎ¹ÎºÎ­Ï‚ Î±Ï€ÏŒ Ï„Î¹Ï‚ Ï€Î¹Î¿
Î²Î±ÏƒÎ¹ÎºÎ­Ï‚ Î¹Î´Î¹ÏŒÏ„Î·Ï„ÎµÏ‚ Ï„Î¿Ï… Î¿Î»Î¿ÎºÎ»Î·ÏÏŽÎ¼Î±Ï„Î¿Ï‚ \textlatin{Riemann}. ÎŸÎ¹ Î±Ï€Î¿Î´ÎµÎ¯Î¾ÎµÎ¹Ï‚
Ï„Ï‰Î½ Ï…Ï€Î¿Î»Î¿Î¯Ï€Ï‰Î½ ÎµÎ¯Î½Î±Î¹ Î¼Î¹Î± ÎºÎ±Î»Î® Î¬ÏƒÎºÎ·ÏƒÎ· Ï€Î¿Ï… Î¸Î± ÏƒÎ±Ï‚ Î²Î¿Î·Î¸Î®ÏƒÎµÎ¹ Î½Î±
ÎµÎ¾Î¿Î¹ÎºÎµÎ¹Ï‰Î¸ÎµÎ¯Ï„Îµ Î¼Îµ Ï„Î¹Ï‚ Î´Î¹Î±Î¼ÎµÏÎ¯ÏƒÎµÎ¹Ï‚, Ï„Î± Î¬Î½Ï‰ ÎºÎ±Î¹ ÎºÎ¬Ï„Ï‰ Î±Î¸ÏÎ¿Î¯ÏƒÎ¼Î±Ï„Î± ÎºÎ»Ï€.

\begin{theorem}
Î‘Î½ $f(x)=c$ Î³Î¹Î± ÎºÎ¬Î¸Îµ $x\in [a,b]$, Ï„ÏŒÏ„Îµ
$$\int_a^bf(x)dx =c(b-a).\leqno (4.4.1)$$
\end{theorem}



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\chapter{Î¤ÎµÏ‡Î½Î¹ÎºÎ­Ï‚ Î¿Î»Î¿ÎºÎ»Î®ÏÏ‰ÏƒÎ·Ï‚}
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Î£Îµ Î±Ï…Ï„ÏŒ Ï„Î¿ ÎšÎµÏ†Î¬Î»Î±Î¹Î¿ Ï€ÎµÏÎ¹Î³ÏÎ¬Ï†Î¿Ï…Î¼Îµ, Ï‡Ï‰ÏÎ¯Ï‚ Î¹Î´Î¹Î±Î¯Ï„ÎµÏÎ· Î±Ï…ÏƒÏ„Î·ÏÏŒÏ„Î·Ï„Î±, Ï„Î¹Ï‚
Î²Î±ÏƒÎ¹ÎºÎ­Ï‚ Î¼ÎµÎ¸ÏŒÎ´Î¿Ï…Ï‚ Ï…Ï€Î¿Î»Î¿Î³Î¹ÏƒÎ¼Î¿Ï Î¿Î»Î¿ÎºÎ»Î·ÏÏ‰Î¼Î¬Ï„Ï‰Î½. Î”Î¯Î½ÎµÏ„Î±Î¹ Î¼Î¹Î± ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ·
$f$ ÎºÎ±Î¹ Î¸Î­Î»Î¿Ï…Î¼Îµ Î½Î± Î²ÏÎ¿ÏÎ¼Îµ Î¼Î¹Î± Î±Î½Ï„Î¹Ï€Î±ÏÎ¬Î³Ï‰Î³Î¿ Ï„Î·Ï‚ $f$, Î´Î·Î»Î±Î´Î® Î¼Î¹Î±
ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ· $F$ Î¼Îµ Ï„Î·Î½ Î¹Î´Î¹ÏŒÏ„Î·Ï„Î± $F^{\prime }=f$. Î¤ÏŒÏ„Îµ,
$$\int f(x)dx =F(x)+c.$$

\section{ÎŸÎ»Î¿ÎºÎ»Î®ÏÏ‰ÏƒÎ· Î¼Îµ Î±Î½Ï„Î¹ÎºÎ±Ï„Î¬ÏƒÏ„Î±ÏƒÎ·}

\subsection{Î&nbsp;Î¯Î½Î±ÎºÎ±Ï‚ ÏƒÏ„Î¿Î¹Ï‡ÎµÎ¹Ï‰Î´ÏŽÎ½ Î¿Î»Î¿ÎºÎ»Î·ÏÏ‰Î¼Î¬Ï„Ï‰Î½}

ÎšÎ¬Î¸Îµ Ï„ÏÏ€Î¿Ï‚ Ï€Î±ÏÎ±Î³ÏŽÎ³Î¹ÏƒÎ·Ï‚ $F^{\prime }(x)=f(x)$ Î¼Î±Ï‚ Î´Î¯Î½ÎµÎ¹ Î­Î½Î±Î½ Ï„ÏÏ€Î¿
Î¿Î»Î¿ÎºÎ»Î®ÏÏ‰ÏƒÎ·Ï‚: Î· $F$ ÎµÎ¯Î½Î±Î¹ Î±Î½Ï„Î¹Ï€Î±ÏÎ¬Î³Ï‰Î³Î¿Ï‚ Ï„Î·Ï‚ $f$. ÎœÏ€Î¿ÏÎ¿ÏÎ¼Îµ Î­Ï„ÏƒÎ¹ Î½Î±
Î´Î·Î¼Î¹Î¿Ï…ÏÎ³Î®ÏƒÎ¿Ï…Î¼Îµ Î­Î½Î±Î½ Ï€Î¯Î½Î±ÎºÎ± Î²Î±ÏƒÎ¹ÎºÏŽÎ½ Î¿Î»Î¿ÎºÎ»Î·ÏÏ‰Î¼Î¬Ï„Ï‰Î½, Î±Î½Ï„Î¹ÏƒÏ„ÏÎ­Ï†Î¿Î½Ï„Î±Ï‚
Ï„Î¿Ï…Ï‚ Ï„ÏÏ€Î¿Ï…Ï‚ Ï€Î±ÏÎ±Î³ÏŽÎ³Î¹ÏƒÎ·Ï‚ Ï„Ï‰Î½ Ï€Î¹Î¿ Î²Î±ÏƒÎ¹ÎºÏŽÎ½ ÏƒÏ…Î½Î±ÏÏ„Î®ÏƒÎµÏ‰Î½:
\begin{eqnarray*}
\int x^adx =\frac{x^{a+1}}{a+1},\qquad a\neq -1, &amp;&amp;
\int\frac{1}{x}\,dx = \ln |x| +c\\
\int e^xdx = e^x+c, &amp;&amp;
\int\sin x\,dx = -\cos x+c\\
\int\cos x\,dx = \sin x+c, &amp;&amp;
\int\frac{1}{\cos^2x}\,dx = \tan x+c\\
\int\frac{1}{\sin^2x}\,dx = -\cot x+c , &amp;&amp;
\int\frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x+c\\
\int\frac{1}{1+x^2}\,dx =\arctan x+c. &amp;&amp;
\end{eqnarray*}


\section{ÎŸÎ»Î¿ÎºÎ»Î®ÏÏ‰ÏƒÎ· ÏÎ·Ï„ÏŽÎ½ ÏƒÏ…Î½Î±ÏÏ„Î®ÏƒÎµÏ‰Î½}

Î£Îµ Î±Ï…Ï„Î® Ï„Î·Î½ Ï€Î±ÏÎ¬Î³ÏÎ±Ï†Î¿ Ï€ÎµÏÎ¹Î³ÏÎ¬Ï†Î¿Ï…Î¼Îµ Î¼Î¹Î± Î¼Î­Î¸Î¿Î´Î¿ Î¼Îµ Ï„Î·Î½ Î¿Ï€Î¿Î¯Î± Î¼Ï€Î¿ÏÎµÎ¯
ÎºÎ±Î½ÎµÎ¯Ï‚ Î½Î± Ï…Ï€Î¿Î»Î¿Î³Î¯ÏƒÎµÎ¹ Ï„Î¿ Î±ÏŒÏÎ¹ÏƒÏ„Î¿ Î¿Î»Î¿ÎºÎ»Î®ÏÏ‰Î¼Î± Î¿Ï€Î¿Î¹Î±ÏƒÎ´Î®Ï€Î¿Ï„Îµ ÏÎ·Ï„Î®Ï‚
ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ·Ï‚
$$f(x)=\frac{p(x)}{q(x)}=\frac{a_nx^n+a_{n-1}x^{n-1}+\cdots
+a_1x+a_0}{b_mx^m+b_{m-1}x^{m-1}+\cdots +b_1x+b_0}.\leqno (6.3.1)$$
Î— Ï€ÏÏŽÏ„Î· Ï€Î±ÏÎ±Ï„Î®ÏÎ·ÏƒÎ· ÎµÎ¯Î½Î±Î¹ ÏŒÏ„Î¹ Î¼Ï€Î¿ÏÎ¿ÏÎ¼Îµ Ï€Î¬Î½Ï„Î± Î½Î± Ï…Ï€Î¿Î¸Î­Ï„Î¿Ï…Î¼Îµ ÏŒÏ„Î¹ $n&lt;m$.
Î‘Î½ Î¿ Î²Î±Î¸Î¼ÏŒÏ‚ $n$ Ï„Î¿Ï… Î±ÏÎ¹Î¸Î¼Î·Ï„Î® $p(x)$ ÎµÎ¯Î½Î±Î¹ Î¼ÎµÎ³Î±Î»ÏÏ„ÎµÏÎ¿Ï‚ Î® Î¯ÏƒÎ¿Ï‚ Î±Ï€ÏŒ Ï„Î¿Î½
Î²Î±Î¸Î¼ÏŒ $m$ Ï„Î¿Ï… Ï€Î±ÏÎ¿Î½Î¿Î¼Î±ÏƒÏ„Î® $q(x)$, Ï„ÏŒÏ„Îµ Î´Î¹Î±Î¹ÏÎ¿ÏÎ¼Îµ Ï„Î¿ $p(x)$ Î¼Îµ Ï„Î¿
$q(x)$: Ï…Ï€Î¬ÏÏ‡Î¿Ï…Î½ Ï€Î¿Î»Ï…ÏŽÎ½Ï…Î¼Î± $\pi (x)$ ÎºÎ±Î¹ $\upsilon (x)$ ÏŽÏƒÏ„Îµ Î¿
Î²Î±Î¸Î¼ÏŒÏ‚ Ï„Î¿Ï… $\upsilon (x)$ Î½Î± ÎµÎ¯Î½Î±Î¹ Î¼Î¹ÎºÏÏŒÏ„ÎµÏÎ¿Ï‚ Î±Ï€ÏŒ $m$ ÎºÎ±Î¹ $$p(x)=\pi
(x)q(x)+\upsilon (x).\leqno (6.3.2)$$ Î¤ÏŒÏ„Îµ,
$$f(x)=\frac{\pi (x)q(x)+\upsilon (x)}{q(x)}=\pi (x)+\frac{\upsilon
(x)}{q(x)}.\leqno (6.3.3)$$ Î£Ï…Î½ÎµÏ€ÏŽÏ‚, Î³Î¹Î± Ï„Î¿Î½ Ï…Ï€Î¿Î»Î¿Î³Î¹ÏƒÎ¼ÏŒ Ï„Î¿Ï… $\int
f(x)\,dx$ Î¼Ï€Î¿ÏÎ¿ÏÎ¼Îµ Ï„ÏŽÏÎ± Î½Î± Ï…Ï€Î¿Î»Î¿Î³Î¯ÏƒÎ¿Ï…Î¼Îµ Ï‡Ï‰ÏÎ¹ÏƒÏ„Î¬ Ï„Î¿ $\int \pi
(x)\,dx$ (Î±Ï€Î»ÏŒ Î¿Î»Î¿ÎºÎ»Î®ÏÏ‰Î¼Î± Ï€Î¿Î»Ï…Ï‰Î½Ï…Î¼Î¹ÎºÎ®Ï‚ ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ·Ï‚) ÎºÎ±Î¹ Ï„Î¿
$\int\frac{\upsilon (x)}{q(x)}\,dx$ (ÏÎ·Ï„Î® ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ· Î¼Îµ Ï„Î·Î½ Ï€ÏÏŒÏƒÎ¸ÎµÏ„Î·
Î¹Î´Î¹ÏŒÏ„Î·Ï„Î± ÏŒÏ„Î¹ $\mathrm{deg}(\upsilon )&lt;\mathrm{deg}(q)$).

Î¥Ï€Î¿Î¸Î­Ï„Î¿Ï…Î¼Îµ Î»Î¿Î¹Ï€ÏŒÎ½ ÏƒÏ„Î· ÏƒÏ…Î½Î­Ï‡ÎµÎ¹Î± ÏŒÏ„Î¹ $f=p/q$ ÎºÎ±Î¹ $\mathrm{deg}(p)&lt;
\mathrm{deg}(q)$. ÎœÏ€Î¿ÏÎ¿ÏÎ¼Îµ ÎµÏ€Î¯ÏƒÎ·Ï‚ Î½Î± Ï…Ï€Î¿Î¸Î­ÏƒÎ¿Ï…Î¼Îµ ÏŒÏ„Î¹ $a_n=b_m=1$.
Î§ÏÎ·ÏƒÎ¹Î¼Î¿Ï€Î¿Î¹Î¿ÏÎ¼Îµ Ï„ÏŽÏÎ± Ï„Î¿ Î³ÎµÎ³Î¿Î½ÏŒÏ‚ ÏŒÏ„Î¹ ÎºÎ¬Î¸Îµ Ï€Î¿Î»Ï…ÏŽÎ½Ï…Î¼Î¿ Î±Î½Î±Î»ÏÎµÏ„Î±Î¹ ÏƒÎµ
Î³Î¹Î½ÏŒÎ¼ÎµÎ½Î¿ Ï€ÏÏ‰Ï„Î¿Î²Î¬Î¸Î¼Î¹Ï‰Î½ ÎºÎ±Î¹ Î´ÎµÏ…Ï„ÎµÏÎ¿Î²Î¬Î¸Î¼Î¹Ï‰Î½ ÏŒÏÏ‰Î½. Î¤Î¿ $q(x)=x^m+\cdots
+b_1x+b_0$ Î³ÏÎ¬Ï†ÎµÏ„Î±Î¹ ÏƒÏ„Î· Î¼Î¿ÏÏ†Î®
$$q(x)=(x-\alpha_1)^{r_1}\cdots
(x-\alpha_k)^{r_k}(x^2+\beta_1x+\gamma_1)^{s_1}\cdots
(x^2+\beta_lx+\gamma_l)^{s_l}.\leqno (6.3.4)$$ ÎŸÎ¹ $\alpha_1,\ldots
,\alpha_k$ ÎµÎ¯Î½Î±Î¹ Î¿Î¹ Ï€ÏÎ±Î³Î¼Î±Ï„Î¹ÎºÎ­Ï‚ ÏÎ¯Î¶ÎµÏ‚ Ï„Î¿Ï… $q(x)$ (ÎºÎ±Î¹ $r_j$ ÎµÎ¯Î½Î±Î¹ Î·
Ï€Î¿Î»Î»Î±Ï€Î»ÏŒÏ„Î·Ï„Î± Ï„Î·Ï‚ ÏÎ¯Î¶Î±Ï‚ $\alpha_j$) ÎµÎ½ÏŽ Î¿Î¹ ÏŒÏÎ¿Î¹
$x^2+\beta_ix+\gamma_i$ ÎµÎ¯Î½Î±Î¹ Ï„Î± Î³Î¹Î½ÏŒÎ¼ÎµÎ½Î±
$(x-z_i)(x-\overline{z_i})$ ÏŒÏ€Î¿Ï… $z_i$ Î¿Î¹ Î¼Î¹Î³Î±Î´Î¹ÎºÎ­Ï‚ ÏÎ¯Î¶ÎµÏ‚ Ï„Î¿Ï… $q(x)$
(ÎºÎ±Î¹ $s_i$ ÎµÎ¯Î½Î±Î¹ Î· Ï€Î¿Î»Î»Î±Ï€Î»ÏŒÏ„Î·Ï„Î± Ï„Î·Ï‚ ÏÎ¯Î¶Î±Ï‚ $z_i$). Î&nbsp;Î±ÏÎ±Ï„Î·ÏÎ®ÏƒÏ„Îµ ÏŒÏ„Î¹
ÎºÎ¬Î¸Îµ ÏŒÏÎ¿Ï‚ Ï„Î·Ï‚ Î¼Î¿ÏÏ†Î®Ï‚ $x^2+\beta_ix+\gamma_i$ Î­Ï‡ÎµÎ¹ Î±ÏÎ½Î·Ï„Î¹ÎºÎ®
Î´Î¹Î±ÎºÏÎ¯Î½Î¿Ï…ÏƒÎ±. Î•Ï€Î¯ÏƒÎ·Ï‚, Î¿Î¹ $k,s\geq 0$ ÎºÎ±Î¹ $r_1+\cdots +r_k+2s_1+\cdots
+2s_l=m$ (Î¿ Î²Î±Î¸Î¼ÏŒÏ‚ Ï„Î¿Ï… $q(x)$).

Î“ÏÎ¬Ï†Î¿Ï…Î¼Îµ Ï„Î·Î½ $f(x)$ ÏƒÏ„Î· Î¼Î¿ÏÏ†Î®
$$f(x)=\frac{x^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0}{(x-\alpha_1)^{r_1}\cdots
(x-\alpha_k)^{r_k}(x^2+\beta_1x+\gamma_1)^{s_1}\cdots
(x^2+\beta_lx+\gamma_l)^{s_l}},\leqno (6.3.5)$$ ÎºÎ±Î¹ Ï„Î·Î½ ((Î±Î½Î±Î»ÏÎ¿Ï…Î¼Îµ
ÏƒÎµ Î±Ï€Î»Î¬ ÎºÎ»Î¬ÏƒÎ¼Î±Ï„Î±)): Ï…Ï€Î¬ÏÏ‡Î¿Ï…Î½ ÏƒÏ…Î½Ï„ÎµÎ»ÎµÏƒÏ„Î­Ï‚ $A_{jt}$,
$B_{it},\Gamma_{it}$ ÏŽÏƒÏ„Îµ
\begin{eqnarray*}
f(x) &amp;=&amp;
\frac{A_{11}}{x-\alpha_1}+\frac{A_{12}}{(x-\alpha_1)^2}+\cdots
+\frac{A_{1r_1}}{(x-\alpha_1)^{r_1}}\\
&amp;&amp; +\cdots \\
&amp;&amp; +\frac{A_{k1}}{x-\alpha_k}+\frac{A_{k2}}{(x-\alpha_k)^2}+\cdots
+\frac{A_{kr_1}}{(x-\alpha_k)^{r_k}}\\
&amp;&amp;
+\frac{B_{11}x+\Gamma_{11}}{x^2+\beta_1x+\gamma_1}+\frac{B_{12}x+\Gamma_{12}}{(x^2+\beta_1x+\gamma_1)^2}+\cdots
+\frac{B_{1s_1}x+\Gamma_{1s_1}}{(x^2+\beta_1x+\gamma_1)^{s_1}}\\
&amp;&amp; +\cdots \\
&amp;&amp;
+\frac{B_{l1}x+\Gamma_{l1}}{x^2+\beta_lx+\gamma_l}+\frac{B_{l2}x+\Gamma_{l2}}{(x^2+\beta_lx+\gamma_l)^2}+\cdots
+\frac{B_{ls_1}x+\Gamma_{ls_l}}{(x^2+\beta_lx+\gamma_l)^{s_l}}.
\end{eqnarray*}


\end{document}
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