%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %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 18:02:48} %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 - Integr\U{e1}lny po\U{10d}et funkci\U{ed} jednej re\U{e1}lnej premennej - Cvi\U{10d}enia\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{Integr\'{a}lny po\v{c}et funkci\'{\i} jednej re\'{a}lnej premennej} \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M7.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O7.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \section{Cvi\v{c}enia} \textbf{Pr\'{\i}klad 1. }Vypo\v{c}\'{\i}tajte $D(f,P)$ a $H(f,P)$ ak $% f:\left\langle -1,2\right\rangle \longrightarrow \mathbf{R,\,}f(x)=x^{4}$ a delenie $P=\{-1,-\frac{1}{2},0,\frac{1}{2},1,\frac{3}{2},2\}$.% \CustomNote{Answer}{% \[ D(f,P)=\left( -\frac{1}{2}\right) ^{4}\left( -\frac{1}{2}+1\right) +0^{4}\left( 0+\frac{1}{2}\right) +0^{4}\left( \frac{1}{2}-0\right) + \]% \[ +\left( \frac{1}{2}\right) ^{4}\left( 1-\frac{1}{2}\right) +1^{4}\left( \frac{3}{2}-1\right) \left( \frac{3}{2}\right) ^{4}\left( 2-\frac{3}{2}% \right) =\allowbreak \frac{85}{64} \]% \par \[ H(f,P)=\left( -1\right) ^{4}\left( -\frac{1}{2}+1\right) +\left( -\frac{1}{2}% \right) ^{4}\left( 0+\frac{1}{2}\right) +\left( \frac{1}{2}\right) ^{4}\left( \frac{1}{2}-0\right) + \]% \[ +\left( 1\right) ^{4}\left( 1-\frac{1}{2}\right) +\left( \frac{3}{2}\right) ^{4}\left( \frac{3}{2}-1\right) \left( 2\right) ^{4}\left( 2-\frac{3}{2}% \right) =\allowbreak \frac{341}{16} \]% } \textbf{Pr\'{\i}klad 2. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \left( 3x^{3}+2x-4\right) dx$.% \CustomNote{Answer}{$\frac{3}{4}x^{4}+x^{2}-4x+C$} \textbf{Pr\'{\i}klad 3. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \left( \frac{1}{3}x^{2}-\frac{1}{5}x\right) dx$.% \CustomNote{Answer}{$\frac{1}{9}x^{3}-\frac{1}{10}x^{2}+C$} \textbf{Pr\'{\i}klad 4. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \left( \sqrt{x^{3}}-\frac{1}{\sqrt{x}}\right) dx$.$% \CustomNote{Answer}{$=\allowbreak \frac{2}{5}x^{\frac{5}{2}}-2x^{\frac{1}{2}% }+C$}$ \textbf{Pr\'{\i}klad 5. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \left( x^{2}-2x+1\right) ^{2}dx$. \CustomNote{Answer}{$\frac{1}{5}x^{5}-x^{4}+2x^{3}-2x^{2}+x+C$} \textbf{Pr\'{\i}klad 6. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{\sqrt{% x^{4}+2+x^{-4}}}{x^{3}}dx$.$% \CustomNote{Answer}{$\ln \left| x\right| -\frac{1}{4x^{4}}+C$}$ \textbf{Pr\'{\i}klad 7. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{x(% \sqrt[3]{x}-x\sqrt[3]{x})}{\sqrt[4]{x}}dx$.$% \CustomNote{Answer}{$\allowbreak -\frac{12}{37}\sqrt[12]{x^{37}}+\frac{12}{25% }\sqrt[12]{x^{25}}+C$}$ \textbf{Pr\'{\i}klad 8. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{% x^{3}-1}{x-1}dx$.$% \CustomNote{Answer}{$\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+x+C$}$ \textbf{Pr\'{\i}klad 9. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int e^{x}a^{x}dx $.$% \CustomNote{Answer}{$\frac{e^{x}a^{x}}{1+\ln a}+C$}$ \textbf{Pr\'{\i}klad 10. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \left( 5\cos x-2x^{5}+\frac{3}{1+x^{2}}\right) dx$.$% \CustomNote{Answer}{$5\sin x-\frac{1}{3}x^{6}+3\limfunc{arctg}x+C$}$ \textbf{Pr\'{\i}klad 11. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \left( 10^{-x}+\frac{x^{2}+2}{x^{2}+1}\right) dx$.$% \CustomNote{Answer}{$-\frac{10^{-x}}{\ln {10}}+x+\limfunc{arctg}x+C$}$ \textbf{Pr\'{\i}klad 12. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \left( 2\sin x-3\cos x\right) dx$.$% \CustomNote{Answer}{$-2\cos x-3\sin x+C$}$ \textbf{Pr\'{\i}klad 13. }N\'{a}jdite stredn\'{u} hodnotu funkcie $% f(x)=2x^{2}+3x+3$ na $\langle 1,4\rangle $.% \CustomNote{Prob_Solv_Hint}{% Stredn\'{a} hodnota je definovan\'{a} $\frac{1}{4-1}\int_{1}^{4}\left( 2x^{2}+3x+3\right) dx=\allowbreak \frac{49}{2}$.} \textbf{Pr\'{\i}klad 14. }Nap\"{a}tie v elektrickom okruhu sa v priebehu jednej min\'{u}ty rovnomerne zv\"{a}\v{c}\v{s}uje od $U_{0}=100$ $\unit{V}$ do $U_{1}=120$ $\unit{V}$. N\'{a}jdite stredn\'{u} hodnotu n\'{a}boja $Q$, ktor\'{y} prejde okruhom za tento \v{c}as, ak odpor okruhu je $R=10$ $\unit{% %TCIMACRO{\U{3a9}}% %BeginExpansion \Omega% %EndExpansion }$. \CustomNote{Prob_Solv_Hint}{% Plat\'{\i}: $U(t)=100+\frac{t}{3}$, kde \v{c}as $t$ meriame v sekund\'{a}ch. Potom $I(t)=\frac{U(t)}{R}=10+\frac{t}{30}$. Plat\'{\i} \v{d}alej $\frac{dQ}{% dt}=I(t)=10+\frac{t}{30}$. Potom stredn\'{a} hodnota $\overline{Q}=\frac{1}{% 60}\int_{0}^{60}\left( 10+\frac{t}{30}\right) dt=11$ $\unit{A}\unit{s}$.} \textbf{Pr\'{\i}klad 15. }Nap\"{a}tie v elektrickej sieti sa rovnomerne zmen% \v{s}uje o $0,4$ $\unit{V}/\unit{s}$. Za\v{c}iato\v{c}n\'{e} nap\"{a}tie je $% 100$ $\unit{V}$. Odpor siete je $R=5$ $\unit{% %TCIMACRO{\U{3a9}}% %BeginExpansion \Omega% %EndExpansion }$. N\'{a}jdite stredn\'{u} hodnotu pr\'{a}ce elektrick\'{e}ho pr\'{u}du po% \v{c}as prvej min\'{u}ty.% \CustomNote{Prob_Solv_Hint}{% Plat\'{\i}: $U(t)=100-0,4t$, kde \v{c}as $t$ meriame v sekund\'{a}ch. Potom $% I(t)=\frac{U(t)}{R}=20+0,08t$. Pr\'{a}ca elektrick\'{e}ho pr\'{u}du je dan% \'{a} vz\v{t}ahom: $A(t)=U(t)I(t)=\left( 100-0,4t\right) \left( 20+0,08t\right) $. Potom stredn\'{a} hodnota $\overline{A}=\frac{1}{60}% \int_{0}^{60}\left( 100-0,4t\right) \left( 20+0,08t\right) dt=\allowbreak 40\,400$ $\unit{W}\unit{s}$.} \textbf{Pr\'{\i}klad 16. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{1}{% \sqrt{3-3x^{2}}}dx$.% \CustomNote{Answer}{$\frac{1}{\sqrt{3}}\arcsin x+C$} \textbf{Pr\'{\i}klad 17. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{% 3.2^{x}-2.3^{x}}{2^{x}}dx\allowbreak $.$% \CustomNote{Answer}{$3x-\frac{3^{x}}{2^{x-1}\left( \ln 3-\ln 2\right) }+C$}$ \textbf{Pr\'{\i}klad 18. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{% 1+\cos ^{2}x}{1+\cos {2x}}dx$.$% \CustomNote{Answer}{$\frac{1}{2}\limfunc{tg}x+\frac{1}{2}x+C$}$ \textbf{Pr\'{\i}klad 19. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{\cos {2x}}{\cos ^{2}x\sin ^{2}x}dx$.$% \CustomNote{Answer}{$-\func{cotg}x-\limfunc{tg}x+C$.}$ \textbf{Pr\'{\i}klad 20. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \limfunc{tg% }{}^{2}xdx$.$% \CustomNote{Answer}{$\limfunc{tg}x-x+C$}$ \textbf{Pr\'{\i}klad 21. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \func{cotg}% ^{2}xdx$.$% \CustomNote{Answer}{$-\func{cotg}x-x+C$}$ \textbf{Pr\'{\i}klad 22. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int 2\sin ^{2}{% \left( \frac{x}{2}\right) }dx$.$% \CustomNote{Answer}{$x-\sin x+C$}$ \textbf{Pr\'{\i}klad 23. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{dx}{% \cos {2x}+\sin ^{2}x}$. $% \CustomNote{Answer}{$\limfunc{tg}x+C$}$ \textbf{Pr\'{\i}klad 24. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{% \left( 1+2x^{2}\right) }{x^{2}\left( 1+x^{2}\right) }dx$.$% \CustomNote{Answer}{$-\frac{1}{x}+\limfunc{arctg}x+C$}$ \textbf{Pr\'{\i}klad 25. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int \frac{% \left( 1+x\right) ^{2}}{x\left( 1+x^{2}\right) }dx$.$% \CustomNote{Answer}{$\ln {|x|}+2\limfunc{arctg}x+C$}$ \textbf{Pr\'{\i}klad 26. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int_{0}^{1}% \frac{1}{1+x^{2}}dx$.$% \CustomNote{Answer}{$\frac{\pi }{4}$}$ \textbf{Pr\'{\i}klad 27. }Vypo\v{c}\'{\i}tajte integr\'{a}l $\int_{\frac{1}{2% }}^{\frac{\sqrt{3}}{2}}\frac{1}{\sqrt{1-x^{2}}}dx$.% \CustomNote{Answer}{$\frac{\pi }{6}$} \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M7.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O7.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za I} \section{Integr\'{a}lny po\v{c}et funkci\'{\i} jednej re\'{a}lnej premennej} \end{document}