%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 16:11:08} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - Met\U{f3}dy integr\U{e1}lneho po\U{10d}tu - \U{160}peci\U{e1}lne integra\U{10d}n\U{e9} met\U{f3}dy\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Met\'{o}dy integr\'{a}lneho po\v{c}tu} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M8.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M8.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M82.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M82.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O8.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O8.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C8.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C8.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} V tejto kapitole zopakujeme zn\'{a}me z\'{a}kladn\'{e} znalosti z t\'{y}chto oblast\'{\i}, ktor\'{e} budeme vyu\v{z}\'{\i}va\v{t}. \subsection{\v{S}peci\'{a}lne integra\v{c}n\'{e} met\'{o}dy} \label{3} \textbf{% %TCIMACRO{% %\hyperref{Integr\'{a}ly typu $\int \sin ^{m}x\cos ^{n}xdx$ }{}{}{M83.tex#1}}% %BeginExpansion \msihyperref{Integr\'{a}ly typu $\int \sin ^{m}x\cos ^{n}xdx$ }{}{}{M83.tex#1}% %EndExpansion } \textbf{% %TCIMACRO{% %\hyperref{Integr\'{a}ly typu $\int \sin ax\cos bxdx,\,\int \sin ax\sin bxdx,\,\int \cos ax\cos bxdx$}{}{}{M83.tex#2}}% %BeginExpansion \msihyperref{Integr\'{a}ly typu $\int \sin ax\cos bxdx,\,\int \sin ax\sin bxdx,\,\int \cos ax\cos bxdx$}{}{}{M83.tex#2}% %EndExpansion } \textbf{% %TCIMACRO{% %\hyperref{Integrovanie racion\'{a}lnych funkci\'{\i}}{}{}{M832.tex#1}}% %BeginExpansion \msihyperref{Integrovanie racion\'{a}lnych funkci\'{\i}}{}{}{M832.tex#1}% %EndExpansion } \textbf{% %TCIMACRO{% %\hyperref{Integrovanie iracion\'{a}lnych funkci\'{\i}}{}{}{M833.tex#1}}% %BeginExpansion \msihyperref{Integrovanie iracion\'{a}lnych funkci\'{\i}}{}{}{M833.tex#1}% %EndExpansion } \textbf{% %TCIMACRO{\hyperref{Eulerove substit\'{u}cie}{}{}{M834.tex#1}}% %BeginExpansion \msihyperref{Eulerove substit\'{u}cie}{}{}{M834.tex#1}% %EndExpansion } \textbf{% %TCIMACRO{% %\hyperref{Trigonometrick\'{e} substit\'{u}cie pre v\'{y}razy $\sqrt{a^{2}-x^{2}},\,\sqrt{a^{2}+x^{2}},\,\sqrt{x^{2}-a^{2}},\,\sqrt{\pm a^{2}\pm b^{2}x^{2}}$}{}{}{M835.tex#1}}% %BeginExpansion \msihyperref{Trigonometrick\'{e} substit\'{u}cie pre v\'{y}razy $\sqrt{a^{2}-x^{2}},\,\sqrt{a^{2}+x^{2}},\,\sqrt{x^{2}-a^{2}},\,\sqrt{\pm a^{2}\pm b^{2}x^{2}}$}{}{}{M835.tex#1}% %EndExpansion } \textbf{% %TCIMACRO{% %\hyperref{Integrovanie trigonometrick\'{y}ch funkci\'{\i}}{}{}{M836.tex#1}}% %BeginExpansion \msihyperref{Integrovanie trigonometrick\'{y}ch funkci\'{\i}}{}{}{M836.tex#1}% %EndExpansion } \textbf{% %TCIMACRO{% %\hyperref{Integrovanie transcendentn\'{y}ch funkci\'{\i}}{}{}{M836.tex#2}}% %BeginExpansion \msihyperref{Integrovanie transcendentn\'{y}ch funkci\'{\i}}{}{}{M836.tex#2}% %EndExpansion } \subsubsection{\label{1}Integr\'{a}ly typu $\protect\int \sin ^{m}x\cos ^{n}xdx$} \begin{center} \begin{tabular}{|c|c|c|} \hline $m,n\in \mathbf{N}$ & Substit\'{u}cia & Vhodn\'{a} identita \\ \hline $n$ - nep\'{a}rne & $u=\sin x$ & $\cos ^{2}x=1-\sin ^{2}x$ \\ \hline $m$ - nep\'{a}rne & $u=\cos x$ & $\sin ^{2}x=1-\cos ^{2}x$ \\ \hline $n$ a $m$ - p\'{a}rne & redukova\v{t} na men\v{s}ie mocniny $n$ a $m$ & $% \begin{tabular}{c} $\sin ^{2}x=\frac{1}{2}\left( 1-\cos 2x\right) $ \\ $\cos ^{2}x=\frac{1}{2}\left( 1+\cos 2x\right) $% \end{tabular}% $ \\ \hline \end{tabular} \end{center} \begin{example} Vypo\v{c}\'{\i}tajte $\int \sin ^{2}x\cos ^{3}xdx$. \end{example} \begin{solution} V tomto pr\'{\i}pade je $n=3$ - nep\'{a}rne, tak polo\v{z}\'{\i}me $u=\sin x$% , t.j. $du=\cos xdx$. Potom \[ \int \sin ^{2}x\cos ^{3}xdx=\int \sin ^{2}x\cos ^{2}x\cos xdx=\int \sin ^{2}x\left( 1-\sin ^{2}x\right) \cos xdx= \]% \[ =\int \left( u^{2}-u^{4}\right) du=\frac{u^{3}}{3}-\frac{u^{5}}{5}+C=\frac{% \sin ^{3}x}{3}-\frac{\sin ^{5}x}{5}+C.\square \] \end{solution} \begin{example} Vypo\v{c}\'{\i}tajte $\int \sin ^{2}x\cos ^{2}xdx$. \end{example} \begin{solution} V tomto pr\'{\i}pade je $m,n$ p\'{a}rne. Podintegr\'{a}lny v\'{y}raz zredukujeme na tvar mocniny funkcie kos\'{\i}nus a potom pou\v{z}ijeme %TCIMACRO{\hyperref{rekurentn\'{e} vz\v{t}ahy}{}{}{M81.tex#3}}% %BeginExpansion \msihyperref{rekurentn\'{e} vz\v{t}ahy}{}{}{M81.tex#3}% %EndExpansion :% \[ \int \sin ^{2}x\cos ^{2}xdx=\frac{1}{4}\int \left( 1-\cos 2x\right) \left( 1+\cos 2x\right) dx=\frac{1}{4}\int \left( 1-\cos ^{2}2x\right) dx= \]% \[ =\frac{1}{4}\int 1dx-\frac{1}{4}\int \cos ^{2}2xdx=\frac{1}{4}x-\frac{1}{4}% \frac{1}{2}\int \left( 1+\cos 4x\right) dx= \]% \[ =\frac{1}{4}x-\frac{1}{8}x-\frac{1}{8}\frac{1}{4}\sin 4x+C=\frac{1}{8}x-% \frac{1}{32}\sin 4x+C.\square \] \end{solution} \begin{center} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{Sp\"{a}\v{t}}{}{}{M83.tex#3}}% %BeginExpansion \msihyperref{Sp\"{a}\v{t}}{}{}{M83.tex#3}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsubsection{\label{2}Integr\'{a}ly typu $\protect\int \sin ax\cos bxdx,\,% \protect\int \sin ax\sin bxdx,\,\protect\int \cos ax\cos bxdx$} V tomto pr\'{\i}pade pou\v{z}ijeme jednu z trigonometrick\'{y}ch ident\'{\i}% t \[ \sin x\cos y=\frac{1}{2}\left[ \sin \left( x-y\right) +\sin \left( x+y\right) \right] , \]% \[ \sin x\sin y=\frac{1}{2}\left[ \cos \left( x-y\right) -\cos \left( x+y\right) \right] , \]% \[ \cos x\cos y=\frac{1}{2}\left[ \cos \left( x-y\right) +\cos \left( x+y\right) \right] . \] \begin{example} Vypo\v{c}\'{\i}tajte $\int \sin 5x\sin 3xdx.$ \end{example} \begin{solution} $\int \sin 5x\sin 3xdx=\frac{1}{2}\int \cos 2xdx-\frac{1}{2}\int \cos 8xdx=% \frac{1}{4}\sin 2x-\frac{1}{16}\sin 8x+C.$ Pou\v{z}ili sme v\'{y}sledok %TCIMACRO{\hyperref{pr\'{\i}kladu}{}{}{M82.tex#5}}% %BeginExpansion \msihyperref{pr\'{\i}kladu}{}{}{M82.tex#5}% %EndExpansion . \end{solution} \begin{center} \begin{tabular}{|c|} \hline %TCIMACRO{\hyperref{{\small Sp\"{a}\v{t}}}{}{}{M83.tex#3} }% %BeginExpansion \msihyperref{{\small Sp\"{a}\v{t}}}{}{}{M83.tex#3} %EndExpansion \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za I} \section{Met\'{o}dy integr\'{a}lneho po\v{c}tu} \end{document}