%% 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 16:29:25} %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 - Zbierka 4\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{Zbierka \'{u}loh} \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{Z.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{Z.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{Integr\'{a}ly.} Vypo\v{c}\'{\i}tajte nasleduj\'{u}ce integr\'{a}ly: \textbf{Pr\'{\i}klad 1. }$\int \frac{2x+1}{x^{2}+2x+5}dx\;% \CustomNote{Answer}{$\ln \left( x^{2}+2x+5\right) -\frac{1}{2}\func{arctg}% \left( \frac{x+1}{2}\right) +C$}$ \textbf{Pr\'{\i}klad 2. }$\int \frac{x+2}{x^{3}+x^{2}+5x-7}dx\;% \CustomNote{Answer}{$\frac{3}{10}\ln \left| x-1\right| -\frac{3}{20}\ln \left( x^{2}+2x+7\right) +\frac{\sqrt{6}}{15}\func{arctg}\left( \frac{\sqrt{6% }\left( x+1\right) }{6}\right) +C$}$ \textbf{Pr\'{\i}klad 3. }$\int \frac{1}{1-x^{4}}dx\;% \CustomNote{Answer}{$\frac{1}{4}\ln \left| x+1\right| -\frac{1}{4}\ln \left| x-1\right| +\frac{1}{2}\func{arctg}x+C.$}$ \textbf{Pr\'{\i}klad 4. }$\int \frac{x}{x^{8}-1}dx\;% \CustomNote{Prob_Solv_Hint}{$\frac{1}{8}\ln \left| \frac{x^{2}-1}{x^{2}+1}% \right| -\frac{1}{4}\func{arctg}\left( x^{2}\right) +C,\,$n\'{a}vod substit% \'{u}cia $t=x^{2}$}$ $\allowbreak $ \textbf{Pr\'{\i}klad 5. }$\int \frac{x^{4}-2x^{3}-x^{2}+3x-1}{x^{2}-2x}dx\;% \CustomNote{Answer}{$\frac{x^{3}}{3}-x+\frac{1}{2}\ln \left| x^{2}-2x\right| +C.$}$ \textbf{Pr\'{\i}klad 6. }$\int \frac{x^{2}+x}{x^{6}+1}dx\;% \CustomNote{Answer}{$\frac{1}{3}\func{arctg}\left( x^{3}\right) +\frac{1}{6}% \ln \left| x^{2}+1\right| -\frac{1}{12}\ln \left| x^{4}-x^{2}+1\right| +% \frac{\sqrt{3}}{6}\func{arctg}\left( \frac{\sqrt{3}\left( 2x^{2}-1\right) }{3% }\right) +C.$}$ \textbf{Pr\'{\i}klad 7. }$\int \frac{7-x}{x^{3}-x^{2}+3x+5}dx\;% \CustomNote{Answer}{$\allowbreak \ln \frac{\left| x+1\right| }{\sqrt{% x^{2}-2x+5}}+\frac{1}{2}\func{arctg}\left( \frac{x-1}{2}\right) +C.$}$ \textbf{Pr\'{\i}klad 8. }$\int \frac{1}{x^{3}+1}dx\;% \CustomNote{Answer}{$\allowbreak \frac{1}{3}\ln \left| x+1\right| -\frac{1}{6% }\ln \left( x^{2}-x+1\right) +\frac{\sqrt{3}}{3}\func{arctg}\frac{\sqrt{3}% \allowbreak }{3}\left( 2x-1\right) +C.$}$ \textbf{Pr\'{\i}klad 9. }$\int \frac{5x^{3}+x^{2}+3x}{\left( x+2\right) \left( x^{2}+x+1\right) \left( x^{2}-x+1\right) }dx\;\allowbreak \CustomNote{Answer}{$\ln \frac{\sqrt{x^{4}+x^{2}+1}}{\left( x+2\right) ^{2}}+% \frac{\sqrt{3}}{3}\func{arctg}\frac{\sqrt{3}}{3}\left( 2x+1\right) +\allowbreak \frac{\sqrt{3}}{3}\func{arctg}\frac{\sqrt{3}}{3}\left( 2x-1\right) +C.$}$ \textbf{Pr\'{\i}klad 10. }$\int \frac{7x-2}{\left( x^{2}-6x+13\right) ^{2}}% dx\;% \CustomNote{Answer}{$\allowbreak \frac{1}{16}\cdot \frac{38x-170}{x^{2}-6x+13% }+\frac{19}{16}\func{arctg}\left( \frac{x-3}{2}\right) +C.$}$ \textbf{Pr\'{\i}klad 11. }$\int \frac{6x-13}{\left( 4x^{2}+4x+17\right) ^{2}}% dx\;% \CustomNote{Answer}{$\allowbreak \frac{1}{2}\cdot \frac{-x-2}{4x^{2}+4x+17}-% \frac{1}{16}\func{arctg}\left( \frac{2x+1}{4}\right) +C.$}$ \textbf{Pr\'{\i}klad 12. }$\int \frac{-3x^{3}-10x^{2}-8x-3}{\left( x+1\right) ^{2}\left( x^{3}-1\right) }dx\;% \CustomNote{Answer}{$\allowbreak -\frac{1}{x+1}+\ln \frac{x^{2}+x+1}{\left( x-1\right) ^{2}}-\frac{2}{3}\sqrt{3}\func{arctg}\frac{\sqrt{3}\allowbreak }{3% }\left( 2x+1\right) +C.$}$ \textbf{Pr\'{\i}klad 13. }$\int \frac{2x^{3}-2x^{2}+4x-4}{x^{4}+4}% dx\;\allowbreak \CustomNote{Prob_Solv_Hint}{$\frac{1}{2}\ln \left[ \left( x^{2}-2x+2\right) \left( x^{2}+2x+2\right) \right] -2\func{arctg}\left( x+1\right) \allowbreak +C,$% \par $\,$n\'{a}vod $x^{4}+4=x^{4}+4x^{2}+4-4x^{2}$}$ \textbf{Pr\'{\i}klad 14. }$\int_{0}^{3}\left| 2-x\right| dx\;% \CustomNote{Answer}{$\allowbreak \frac{5}{2}.$}$ \textbf{Pr\'{\i}klad 15. }$\int \frac{1}{2^{x}+3}dx\;% \CustomNote{Answer}{$\frac{1}{3\ln 2}\ln \frac{2^{x}}{\left( 2^{x}+3\right) }% +C.$}$ \textbf{Pr\'{\i}klad 16. }$\int \frac{e^{2x}+e^{x}}{e^{2x}+1}dx\;% \CustomNote{Answer}{$\frac{1}{2}\ln \left( e^{2x}+1\right) +\func{arctg}% \left( e^{x}\right) +C.$}$ \textbf{Pr\'{\i}klad 17. }$\int \frac{e^{3x}-e^{x}}{e^{2x}-2e^{x}+1}dx\;% \CustomNote{Answer}{$e^{x}+\ln \left( e^{x}-1\right) ^{2}+C.$}$ \textbf{Pr\'{\i}klad 18. }$\int \frac{e^{x}+10}{\left( e^{2x}-2e^{x}+5\right) }dx\;% \CustomNote{Answer}{$2x-\ln \left| e^{2x}-2e^{x}+5\right| +\frac{3}{2}\func{% arctg}\left( \frac{e^{x}-1}{2}\right) +C.$}$ \textbf{Pr\'{\i}klad 19. }$\int \frac{-e^{4x}+2e^{2x}+e^{x}}{\left( e^{x}+1\right) ^{2}\left( e^{2x}+e^{x}+1\right) }dx\;% \CustomNote{Answer}{$-\frac{1}{e^{x}+1}-\ln \left( e^{x}+1\right) +\frac{2% \sqrt{3}}{3}\func{arctg}\left( \frac{\sqrt{3}\left( 2e^{x}+1\right) }{3}% \right) +C.$}$ \textbf{Pr\'{\i}klad 20. }$\int \frac{\sqrt{x}+\sqrt[4]{x}+\sqrt[3]{x}}{% 2\left( x+\sqrt[6]{x^{7}}\right) }dx\;% \CustomNote{Answer}{$\frac{3}{2}\sqrt[3]{x}+6\sqrt[12]{x}-6\func{arctg}% \left( \sqrt[12]{x}\right) +C.$}$ \textbf{Pr\'{\i}klad 21. }$\int \frac{\sqrt[6]{x}+1}{\sqrt[6]{x^{7}}+\sqrt[4]% {x^{5}}}dx\;% \CustomNote{Answer}{$\frac{12}{\sqrt[12]{x}}-\frac{6}{\sqrt[6]{x}}+24\ln \left| \frac{\sqrt[12]{x}}{\sqrt[12]{x}+1}\right| +C.$}$ \textbf{Pr\'{\i}klad 22. }$\int \frac{\sqrt[3]{x}}{x+\sqrt[6]{x^{5}}}dx\;% \CustomNote{Answer}{$6\left( \frac{\sqrt[3]{x}}{2}-\sqrt[6]{x}+\ln \left( \sqrt[6]{x}+1\right) \right) +C.$}$ \textbf{Pr\'{\i}klad 23. }$\int \sqrt[3]{\frac{x+1}{x-1}}dx\;% \CustomNote{Prob_Solv_Hint}{$% \begin{array}{c} -\frac{2t}{1-t^{2}}+\frac{2}{3}\ln \left| 1-t\right| +\frac{1}{3}\ln \left| t^{2}+t+1\right| +\frac{2\sqrt{3}}{3}\func{arctg}\left( \frac{\sqrt{3}\left( 2t+1\right) }{3}\right) +C,\, \\ \text{n\'{a}vod substit\'{u}cia }\,t=\sqrt[3]{\frac{x+1}{x-1}}% \end{array}% $}$ \textbf{Pr\'{\i}klad 24. }$\int \frac{1-\sqrt[6]{x+1}}{x+1+\sqrt[3]{\left( x+1\right) ^{4}}}dx\;% \CustomNote{Answer}{$\ln \left| x+1\right| -3\ln \left| \sqrt[3]{x+1}% +1\right| -6\func{arctg}\left( \sqrt[6]{x+1}\right) +C.$}$ \textbf{Pr\'{\i}klad 25. }$\int \frac{1}{x+\sqrt{2x-1}}dx\;% \CustomNote{Answer}{$\allowbreak \ln \left( 2x+2\sqrt{2x-1}\right) +\frac{2}{% 1+\sqrt{2x-1}}+C\allowbreak .$}$ \textbf{Pr\'{\i}klad 26. }$\int \frac{1}{x\sqrt[3]{x+1}}dx\;% \CustomNote{Answer}{$\allowbreak \ln \frac{\left| \sqrt[3]{x+1}-1\right| }{% \sqrt{\sqrt[3]{\left( x+1\right) ^{2}}+\sqrt[3]{x+1}+1}}+\allowbreak \sqrt{3}% \func{arctg}\frac{\sqrt{3}}{3}\left( 2\sqrt[3]{x+1}+1\right) +C.$}$ \textbf{Pr\'{\i}klad 27. }$\int \sqrt{\frac{1-x}{1+x}}\frac{dx}{x}\;% \CustomNote{Answer}{$\ln \left| \frac{\sqrt{\frac{1-x}{1+x}}-1}{\sqrt{\frac{% 1-x}{1+x}}+1}\right| +2\func{arctg}\left( \sqrt{\frac{1-x}{1+x}}\right) +C.$} $ \textbf{Pr\'{\i}klad 28. }$\int \frac{\sqrt{x-1}}{x+2\sqrt{x-1}}dx\;% \CustomNote{Answer}{$\allowbreak 2\sqrt{x-1}-\ln \left( 1+\sqrt{x-1}\right) ^{4}-\frac{2}{1+\sqrt{x-1}}+C\allowbreak $}$ \textbf{Pr\'{\i}klad 29. }$\int \frac{1}{x^{2}}\sqrt{\frac{2x+1}{x+1}}dx\;% \CustomNote{Answer}{$\frac{1}{2}\ln \left| \frac{t-1}{t+1}\right| -\frac{1}{% t+1}-\frac{1}{t-1}+C,\,t=\sqrt{\frac{2x+1}{x+1}}.$}$ \textbf{Pr\'{\i}klad 30. }$\int \frac{x^{5}}{\sqrt{1-x^{2}}}dx\;% \CustomNote{Answer}{$\allowbreak -\frac{1}{15}\sqrt{1-x^{2}}\left( 3x^{4}+4x^{2}+8\allowbreak \right) +C.$}$ \textbf{Pr\'{\i}klad 31. }$\int \frac{1}{\sqrt{1+x-2x^{2}}}dx\;% \CustomNote{Answer}{$\frac{\sqrt{2}}{2}\left( \arcsin \left( \frac{4}{3}x-% \frac{1}{3}\right) \right) \allowbreak \allowbreak +C.$}$ \textbf{Pr\'{\i}klad 32. }$\int \frac{1}{\sqrt{8-6x-9x^{2}}}dx\;% \CustomNote{Answer}{$\frac{1}{3}\left( \arcsin \left( x+\frac{1}{3}\right) \right) \allowbreak \allowbreak +C.$}$ \textbf{Pr\'{\i}klad 33. }$\int \frac{1}{\sqrt{x\left( 1-x\right) }}dx\;% \CustomNote{Answer}{$\allowbreak \arcsin \left( 2x-1\right) +C.$}$ \textbf{Pr\'{\i}klad 34. }$\int \sqrt{3-2x-x^{2}}dx\;% \CustomNote{Answer}{$\frac{x+1}{2}\sqrt{\left( 3-2x-x^{2}\right) }+2\arcsin \left( \frac{x+1}{2}\right) \allowbreak +C.$}$ \textbf{Pr\'{\i}klad 35. }$\int \frac{1}{\sqrt{9-4x^{2}}}dx\;% \CustomNote{Answer}{$\frac{1}{2}\left( \arcsin \frac{2}{3}x\right) \allowbreak \allowbreak +C.$}$ \textbf{Pr\'{\i}klad 36. }$\int \frac{2x-10}{\sqrt{1+x-x^{2}}}dx\;% \CustomNote{Answer}{$\allowbreak -2\sqrt{\left( 1+x-x^{2}\right) }-9\arcsin \frac{\sqrt{5}\left( 2x-1\right) }{5}+C.$}$ Pou\v{z}it\'{\i}m Eulerovej substit\'{u}cie vypo\v{c}\'{\i}tajte integr\'{a}% ly v pr\'{\i}kladoch 37 - 41: \textbf{Pr\'{\i}klad 37. }$\int \sqrt{x^{2}+4x+3}dx\;% \CustomNote{Answer}{$% \begin{array}{c} -\frac{1}{8}\left( \sqrt{x^{2}+4x+3}-x-2\right) ^{2}+ \\ +\frac{1}{2}\ln \left| \sqrt{\left( x^{2}+4x+3\right) }\allowbreak \allowbreak -x-2\right| + \\ +\frac{1}{8}\frac{1}{\left( \sqrt{x^{2}+4x+3}-x-2\right) ^{2}}+C% \end{array}% $}$ \textbf{Pr\'{\i}klad 38. }$\int_{0}^{1}\frac{1}{\sqrt{x^{2}+3x+1}}dx\;% \CustomNote{Answer}{$\allowbreak \ln \frac{5+2\sqrt{5}}{5}.$}$ \textbf{Pr\'{\i}klad 39. }$\int \frac{1}{x\sqrt{x^{2}+x+1}}dx\;% \CustomNote{Answer}{$\allowbreak \ln \left| \frac{-x-1+\sqrt{x^{2}+x+1}}{1-x-% \sqrt{x^{2}+x+1}}\right| +C.$}$ \textbf{Pr\'{\i}klad 40. }$\int \frac{1}{x-\sqrt{x^{2}-x+1}}dx\;% \CustomNote{Prob_Solv_Hint}{$\allowbreak 2\ln \left| t\right| -\frac{3}{2}% \ln \left| 2t+1\right| +\frac{3}{2}\cdot \frac{1}{2t+1}+C,\,$% \par n\'{a}vod $\ t=\sqrt{x^{2}-x+1}-x$}$ \textbf{Pr\'{\i}klad 41. }$\int \frac{1}{x\sqrt{x^{2}+1}}dx\;% \CustomNote{Answer}{$\ln \left| \frac{\sqrt{x^{2}+1}+x-1}{\sqrt{x^{2}+1}+x+1}% \right| +C.$}$ Vypo\v{c}\'{\i}tajte met\'{o}dou per partes \textbf{Pr\'{\i}klad 42. }$\int x\func{arctg}xdx\;\allowbreak \CustomNote{Answer}{$\frac{1}{2}x^{2}\func{arctg}x-\frac{1}{2}x+\frac{1}{2}% \func{arctg}x+C.$}$ \textbf{Pr\'{\i}klad 43. }$\int x\limfunc{tg}^{2}xdx\;% \CustomNote{Answer}{$\allowbreak -\frac{1}{2}x^{2}+x\limfunc{tg}x+\ln \left| \cos x\right| +C.$}$ \textbf{Pr\'{\i}klad 44. }$\int \cos \left( \ln x\right) dx\;% \CustomNote{Answer}{$\allowbreak \frac{x}{2}\left( \cos \left( \ln x\right) +\sin \left( \ln x\right) \right) +C.$}$ \textbf{Pr\'{\i}klad 45. }$\int_{0}^{\pi }x^{2}\sin xdx\;% \CustomNote{Answer}{$\allowbreak \pi ^{2}-4.$}$ \textbf{Pr\'{\i}klad 46. }$\int_{0}^{\frac{\pi }{2}}\exp \left( 2x\right) \sin xdx\;% \CustomNote{Answer}{$\allowbreak \frac{2}{5}e^{\pi }+\frac{1}{5}.$}$ \textbf{Pr\'{\i}klad 47. }$\int \func{arctg}\frac{1}{x-1}dx\;\allowbreak \CustomNote{Answer}{$x\func{arctg}\frac{1}{x-1}+\frac{1}{2}\ln \left| x^{2}-2x+2\right| +$% \par $+\func{arctg}\left( x-1\right) +C\allowbreak .$}$ \textbf{Pr\'{\i}klad 48. }$\int x^{2}\arcsin xdx\;% \CustomNote{Answer}{$\frac{1}{3}x^{3}\arcsin x+\frac{1}{9}x^{2}\sqrt{\left( 1-x^{2}\right) }+$% \par $+\frac{2}{9}\sqrt{\left( 1-x^{2}\right) }\allowbreak +C.$}$ \textbf{Pr\'{\i}klad 49. }$\int x^{2}\exp \left( \sqrt{x}\right) dx\;% \CustomNote{Answer}{$\allowbreak 2x^{\frac{5}{2}}e^{\sqrt{x}}-10x^{2}e^{% \sqrt{x}}+40x^{\frac{3}{2}}e^{\sqrt{x}}-120+\allowbreak +xe^{\sqrt{x}}+240% \sqrt{x}e^{\sqrt{x}}-240e^{\sqrt{x}}+C.$}$ \textbf{Pr\'{\i}klad 50. }$\int \left( x^{2}+x\right) \ln \left( x+1\right) dx\;% \CustomNote{Prob_Solv_Hint}{$% \begin{array}{c} \allowbreak \frac{2\left( x+1\right) ^{3}-3\left( x+1\right) ^{2}}{6}\ln \left( x+1\right) -\frac{4\left( x+1\right) ^{3}-9\left( x+1\right) ^{2}}{36}% +C,\, \\ \text{n\'{a}vod substit\'{u}cia }\,t=x+1% \end{array}% $}$ \textbf{Pr\'{\i}klad 51. }$\int_{0}^{\frac{\pi }{4}}\exp \left( 3x\right) \sin \left( 4x\right) dx\;% \CustomNote{Answer}{$\allowbreak \frac{4}{25}\left( e^{\frac{3}{4}\pi }+1\right) .$}$ \textbf{Pr\'{\i}klad 52. }$\int \frac{\ln \left( 1-x+x^{2}\right) }{x^{2}}% dx\;\allowbreak \CustomNote{Answer}{$-\frac{1}{x}\ln \left( x^{2}-x+1\right) -\ln \left| x\right| +\frac{1}{2}\ln \left( x^{2}-x+1\right) +$% \par $+\sqrt{3}\func{arctg}\frac{\sqrt{3}\allowbreak }{3}\left( 2x-1\right) +C.$}$ \textbf{Pr\'{\i}klad 53. }$\int \arcsin \left( \sqrt{\frac{x}{x+1}}\right) dx\;\allowbreak \CustomNote{Answer}{$x\arcsin \sqrt{\frac{x}{x+1}}-\sqrt{x}+\func{arctg}% \sqrt{x}+C.$}$ \textbf{Pr\'{\i}klad 54. }$\int \sin x\ln \left( \limfunc{tg}x\right) dx\;% \CustomNote{Answer}{$-\cos x\ln \left( \limfunc{tg}x\right) +\ln \left| \limfunc{tg}\frac{x}{2}\right| +C.$}$ \textbf{Pr\'{\i}klad 55. }$\int \frac{1}{x^{2}}\ln \left( \frac{x+2}{x-2}% \right) dx\;% \CustomNote{Answer}{$\frac{1}{2}\allowbreak \ln \left( \frac{x-2}{x+2}% \right) +\ln \left( \frac{\left| x\right| }{\sqrt{x^{2}-4}}\right) +C.$}$ \textbf{Pr\'{\i}klad 56. }$\int \left( x^{2}+1\right) \arcsin xdx\;% \CustomNote{Answer}{$\allowbreak \frac{1}{3}\left( x^{3}+3x\right) \arcsin x+% \frac{1}{9}\left( x^{2}+11\right) \sqrt{\left( 1-x^{2}\right) }+\allowbreak +C.$}$ \textbf{Pr\'{\i}klad 57. }$\int_{0}^{\frac{\sqrt{2}}{2}}\frac{\arcsin x}{% \sqrt{1+x}}dx\;% \CustomNote{Answer}{$\allowbreak \frac{\pi }{2}\sqrt{\frac{2+\sqrt{2}}{2}}+4% \sqrt{\frac{2-\sqrt{2}}{2}}-4.$}$ \textbf{Pr\'{\i}klad 58. }$\int \frac{\arccos x-x}{\sqrt{1-x^{2}}}dx\;% \CustomNote{Prob_Solv_Hint}{$\allowbreak -\frac{\left( \arccos x\right) ^{2}% }{2}+\sqrt{1-x^{2}}+C,$% \par $\,$n\'{a}vod rozde\v{l}te na dva integr\'{a}ly.}$ \textbf{Pr\'{\i}klad 59. }$\int \frac{\exp \left( \func{arctg}x\right) }{% \sqrt{\left( 1+x^{2}\right) ^{3}}}dx\;% \CustomNote{Prob_Solv_Hint}{$\frac{\left( 1+x\right) \exp \left( \func{arctg}% x\right) }{2\sqrt{1+x^{2}}}+C,\,$% \par n\'{a}vod vo\v{l}te $\,g^{\prime }\left( x\right) =\frac{\exp \left( \func{% arctg}x\right) }{1+x^{2}}.$}$ V pr\'{\i}kladoch 60 - 73 vypo\v{c}\'{\i}tajte integr\'{a}ly z goniometrick% \'{y}ch funkci\'{\i} \textbf{Pr\'{\i}klad 60. }$\int \frac{1}{3-\cos x}dx\;% \CustomNote{Answer}{$\allowbreak \frac{\sqrt{2}}{2}\func{arctg}\left( \sqrt{2% }\limfunc{tg}\frac{x}{2}\right) +C.$}$ \textbf{Pr\'{\i}klad 61. }$\int \frac{1}{3-\cos x}dx\;% \CustomNote{Answer}{$\allowbreak \frac{\sqrt{2}}{2}\func{arctg}\left( \sqrt{2% }\limfunc{tg}\frac{x}{2}\right) +C.$}$ \textbf{Pr\'{\i}klad 62. }$\int_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{1}{% 3+\cos x}dx\;% \CustomNote{Answer}{$\allowbreak \frac{\sqrt{2}}{2}\left( \func{arctg}\frac{% \sqrt{2}}{2}-\func{arctg}\frac{\sqrt{6}}{6}\right) .$}$ \textbf{Pr\'{\i}klad 63. }$\int \frac{1}{\sin ^{3}x}dx\;% \CustomNote{Answer}{$-\frac{1}{8}\frac{1}{\limfunc{tg}^{2}\left( \frac{x}{2}% \right) }+\frac{1}{2}\ln \left| \limfunc{tg}\left( \frac{x}{2}\right) \right| +\frac{1}{8}\limfunc{tg}^{2}\left( \frac{x}{2}\right) +C.$}$ \textbf{Pr\'{\i}klad 64. }$\int \sin ^{2}x\cos ^{6}xdx\;% \CustomNote{Answer}{$% \begin{array}{c} \allowbreak -\frac{1}{8}\sin x\cos ^{7}x+\frac{1}{48}\cos ^{5}x\sin x+ \\ +\frac{5}{192}\cos ^{3}x\sin x+\frac{5}{128}\cos x\sin x+\allowbreak \frac{5% }{128}x+C,\, \\ \text{alebo }\frac{5x}{8}-\frac{\sin 4x}{8}+\frac{\sin ^{3}2x}{3}-\frac{\sin 8x}{64}+C.% \end{array}% $}$ \textbf{Pr\'{\i}klad 65. }$\int \frac{1}{2\sin x-\cos x+5}dx\;% \CustomNote{Answer}{$\allowbreak \frac{\sqrt{5}}{5}\func{arctg}\frac{\sqrt{5}% }{5}\left( 3\limfunc{tg}\left( \frac{x}{2}\right) +1\right) +C.$}$ \textbf{Pr\'{\i}klad 66. }$\int \frac{1}{3+\cos x+\sin x}dx\;% \CustomNote{Answer}{$\frac{2\sqrt{7}}{7}\left( \func{arctg}\frac{\sqrt{7}% \left( 2\limfunc{tg}\left( \frac{x}{2}\right) +1\right) }{7}\right) +C.$}$ \textbf{Pr\'{\i}klad 67. }$\int \cos \left( 3x\right) \cos \left( 4x\right) dx\;% \CustomNote{Answer}{$\allowbreak \frac{1}{2}\sin x+\frac{1}{14}\sin 7x+C.$}$ \textbf{Pr\'{\i}klad 68. }$\int \frac{1-\sin x}{1+\cos x}dx\;% \CustomNote{Answer}{$\allowbreak \limfunc{tg}\frac{x}{2}-\ln \left( \limfunc{% tg}^{2}\left( \frac{x}{2}\right) +1\right) +C.$}$ \textbf{Pr\'{\i}klad 69. }$\int \frac{\sin x\cos x}{\sin ^{2}x+\sin x+3}dx\;% \CustomNote{Answer}{$\frac{1}{2}\ln \left( \sin ^{2}x+\sin x+3\right) -$% \par $-\frac{\sqrt{11}}{11}\func{arctg}\left( \frac{\sqrt{11}}{11}\left( 2\sin x+1\right) \allowbreak \right) +C.$}$ \textbf{Pr\'{\i}klad 70. }$\int \frac{\limfunc{tg}x\cos ^{6}x}{\sin ^{4}x}% dx\;% \CustomNote{Answer}{$\allowbreak -\frac{1}{2\sin ^{2}x}-2\ln \left| \sin x\right| +\frac{\sin ^{2}x}{2}+C.$}$ \textbf{Pr\'{\i}klad 71. }$\int \frac{\left( 3\sin ^{2}x-5\sin x-5\right) \cos x}{\left( \sin x-3\right) \left( \sin ^{2}x-\sin x+1\right) }dx\;% \CustomNote{Answer}{$\ln \left| \sin x-3\right| +\ln \left( \sin ^{2}x-\sin x+1\right) +$% \par $+2\sqrt{3}\func{arctg}\left( \frac{\sqrt{3}\left( 2\sin x-1\right) }{3}% \right) +C.$}$ \textbf{Pr\'{\i}klad 72. }$\int_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{1}{% \cos x-2\sin x+3}dx\;% \CustomNote{Answer}{$\allowbreak -\func{arctg}\left( \frac{\sqrt{3}}{3}% -1\right) .$}$ \textbf{Pr\'{\i}klad 73. }$\int_{\frac{\pi }{4}}^{\frac{\pi }{3}}\frac{1+% \limfunc{tg}^{2}x}{\left( 1+\limfunc{tg}x\right) ^{2}}dx\;% \CustomNote{Answer}{$\allowbreak \frac{2-\sqrt{3}}{2}.$}$ \textbf{Pr\'{\i}klad 74. }$\int_{0}^{\frac{\pi }{2}}\frac{\cos x}{\sin ^{2}x+\sin x-6}dx\;% \CustomNote{Answer}{$\frac{1}{5}\ln \frac{3}{8}.$}$ Vypo\v{c}\'{\i}tajte plo\v{s}n\'{y} obsah rovinnej oblasti ohrani\v{c}enej krivkami: \textbf{Pr\'{\i}klad 75. }$y=x\ln x,\,x=\frac{1}{2},\,x=2,\,y=0.\;% \CustomNote{Answer}{$-\frac{9}{16}+\frac{15}{8}\ln 2.$}$ \textbf{Pr\'{\i}klad 76. }$y=x-1,y^{2}=2x+1.\;% \CustomNote{Answer}{$\frac{16}{3}.$}$ \textbf{Pr\'{\i}klad 77. }$y=x^{2},\,y=\frac{1}{2}x^{2},\,y=2.\;% \CustomNote{Answer}{$\frac{8}{3}\left( 2-\sqrt{2}\right) .$}$ \textbf{Pr\'{\i}klad 78. }$y=\frac{27}{x^{2}+9},\,y=\frac{x^{2}}{6}.\;% \CustomNote{Answer}{$\frac{3}{2}\left( 3\pi -2\right) .$}$ \textbf{Pr\'{\i}klad 79. }$y=\ln x,\,\,y=\ln ^{2}x.\;% \CustomNote{Answer}{$3-e.$}$ \textbf{Pr\'{\i}klad 80. }Parabolou $y=x^{2}-6x+8$ a jej doty\v{c}nicami v bodoch $A=\left[ 1,3\right] ,\,B=\left[ 4,0\right] .\;% \CustomNote{Answer}{$\frac{9}{4}.$}$ \textbf{Pr\'{\i}klad 81. }Kruh $x^{2}+y^{2}=8$ je rozdelen\'{y} parabolou $y=% \frac{x^{2}}{2}$ na dve \v{c}asti. Vypo\v{c}\'{\i}tajte obsah men\v{s}ej z nich. $% \CustomNote{Answer}{$2\pi +\frac{4}{3}.$}$ \textbf{Pr\'{\i}klad 82. }Vypo\v{c}\'{\i}tajte objem zrezan\'{e}ho ku\v{z}e% \v{l}a, ktor\'{y} vznikne rot\'{a}ciou element\'{a}rnej oblasti okolo osi $% o_{x}.$ Polomery jeho podst\'{a}v s\'{u} $r=1,\,R=2$ a v\'{y}\v{s}ka $v=3.\;% \CustomNote{Answer}{$7\pi .$}$ V pr\'{\i}kladoch 83 - 84 $\ $vypo\v{c}\'{\i}tajte objem telesa, ktor\'{e} vznikne rot\'{a}ciou oblasti okolo osi $o_{x}.$ \textbf{Pr\'{\i}klad 83. }Oblas\v{t} je ur\v{c}en\'{a} \v{c}iarami $y=\frac{% 2x}{\pi },\,y=\sin x,\,x\geq 0.\;% \CustomNote{Answer}{$\frac{\pi ^{2}}{12}.$}$ \textbf{Pr\'{\i}klad 84. }Oblas\v{t} je ur\v{c}en\'{a} \v{c}iarami $y=\sin x,\,y=\frac{2x}{\pi }.\;% \CustomNote{Answer}{$\frac{\pi ^{2}}{6}.$}$ \textbf{Pr\'{\i}klad 85. }Vypo\v{c}\'{\i}tajte potenci\'{a}l elektrick\'{e}% ho po\v{l}a vo vzdialenosti $a=20\unit{cm}$ meranej kolmo od stredu vodivej kruhovej dosky $R=10\unit{cm}$ nabitej elektrick\'{y}m n\'{a}bojom $% Q=1.10^{-6}\unit{C}.$ V okol\'{\i} kruhovej dosky je v\'{a}kuum. $% \CustomNote{Answer}{$\varphi =\frac{Q}{2\pi \varepsilon _{0}R^{2}}\left( \sqrt{a^{2}+R^{2}}-a\right) .$}$ \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{Z.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{Z.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za I} \section{Zbierka \'{u}loh} \end{document}