Součásti dokumentu Matematika2Priklady
Zdrojový kód
%\wikiskriptum{Matematika2Priklady}
\section{Pokročilé techniky integrace a zobecněný Riemannův integrál}
% \subsection*{\fbox{Rozcvička}}
%
% \begin{itemize}
% \item
% \begin{priklad}
% \int \frac{\ud x}{x^2-4}
% \end{priklad}
% \res{$\frac{1}{4} \ln \left | \frac{x-2}{x+2} \right|+ C$}
%
% \end{itemize}
%\item \begin{priklad}
% \int \frac{\ud x}{x^4+4}
%\end{priklad}
%\res{$\frac{1}{16} \ln \left ( \frac{x^2+2x+2}{x^2-2x+2} \right)+ \frac{1}{8} \arctg(x+1) + \frac{1}{8} \arctg(x-1) + C$}
\subsection*{\fbox{Zkouškové příklady}}
\begin{enumerate}
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\odstavec{Racionální funkce}
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\item
\begin{priklad}
\int \frac{\ud x}{x^4-1}
\end{priklad}
\res{$-\frac14\ln|x+1|\frac14\ln|x-1|-\frac12\arctg{x} + C$}
\item \begin{priklad}
\int \frac{2x^2+3}{x(x-1)^2} \ud x
\end{priklad}
\res{$3 \ln |x| - \ln |x-1| - \frac{5}{x-1} + C$}
\item \begin{priklad}
\int \frac{-2x}{(x+1)(x^2+1)} \ud x
\end{priklad}
\res{$\ln |x+1| - \frac{1}{2} \ln (x^2 + 1) - \arctg x + C$}
\item \begin{priklad}
\int \frac{\ud x}{x(x^2+x+1)}
\end{priklad}
\res{$\ln |x| - \frac{1}{2}\ln(x^2+x+1)- \frac{1}{\sqrt{3}} \arctg \left (\frac{2}{\sqrt{3}}(x + \frac{1}{2}) \right )+ C$}
\item \begin{priklad}
\int \frac{3x^4+x^3+20x^2+3x+31}{(x+1)(x^2+4)^2} \ud x
\end{priklad}
\res{$2 \ln|x+1| + \frac{1}{2} \ln (x^2+4) - \frac{1}{8} \frac{x}{x^2+4} - \frac{1}{16} \arctg \frac{x}{2} + C$}
\item \begin{priklad}
\int \frac{x^5+2}{x^2-1} \ud x
\end{priklad}
\res{$\frac{1}{4} x^4 + \frac{1}{2} x^2 - \frac{1}{2} \ln |x+1| + \frac{3}{2} \ln |x-1| + C$}
\item \begin{priklad}
\int \frac{x}{(x+1)(x+2)(x+3)} \ud x
\end{priklad}
\res{$2\ln|x+2| -\frac32\ln|x+3|-\frac12\ln|x+1|+C$}
\item \begin{priklad}
\int \frac{2x^2+3}{x^2(x-1)} \ud x
\end{priklad}
\res{$5 \ln |x-1| - 3 \ln |x| + \frac{3}{x} + C$}
\item \begin{priklad}
\int \frac{x^5}{(x-2)^2} \ud x
\end{priklad}
\res{$\frac{1}{4}x^4 + \frac{4}{3}x^3 + 6x^2 + 32x - \frac{32}{x-2} + 80 \ln |x-2| +C$}
\item \begin{priklad}
\int \frac{x+3}{x^2-3x+2} \ud x
\end{priklad}
\res{$5 \ln |x-2| - 4 \ln |x-1| + C$}
\item \begin{priklad}
\int \frac{x^2}{(x-1)^2(x+1)} \ud x
\end{priklad}
\res{$ \frac{3}{4} \ln |x-1| - \frac{1}{2(x-1)} + \frac{1}{4} \ln |x+1| + C$}
\item \begin{priklad}
\int \frac{\ud x}{x^4 - 16}
\end{priklad}
\res{$\frac{1}{32} \ln \left | \frac{x-2}{x+2} \right|- \frac{1}{16} \arctg \frac{x}{2} + C$}
\item \begin{priklad}
\int \frac{\ud x}{(x^2 + 16)^2}
\end{priklad}
\res{$\frac{1}{32}\frac{x}{x^2+16}+\frac{1}{128}\arctg\frac{x}{4}$}
\item \begin{priklad}
\int \frac{\ud x}{x^3+1}
\end{priklad}
\res{$\frac{1}{3} \left ( \ln |x+1| - \frac{1}{2} \ln(x^2-x+1) + \frac{1}{\sqrt{3}} \arctg \left ( \frac{2x-1}{\sqrt{3}} \right ) \right )+C$}
\item \begin{priklad}
\int \frac{x}{x^3+1} \ud x
\end{priklad}
\res{$ \frac{1}{3} \left ( -\ln |x+1| + \frac{1}{2} \ln (x^2-x+1) \sqrt{3} \arctg \frac{2x-1}{\sqrt{3}}\right )+C$}
\item \begin{priklad}
\int \frac{\ud x}{(x^2+1)^3}
\end{priklad}
\res{$\frac{x}{4(1+x^2)^2} + \frac{3x}{8(1+x^2)}+ \frac{3}{8} \arctg x + C$}
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\odstavec{Pokročilé techniky integrace}
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\item
\begin{priklad}
\int \sin^3 x \ud x
\end{priklad}
\res{$\frac{1}{3} \cos^3 x - \cos x + C$}
\item
\begin{priklad}
\int \sin^2 3x \ud x
\end{priklad}
\res{$\frac{1}{2} x - \frac{1}{12} \sin 6x + C$}
\item
\begin{priklad}
\int \cos^4 x \sin^3 x \ud x
\end{priklad}
\res{$-\frac{1}{5} \cos^5 x + \frac{1}{7}\cos^7 x + C$}
\item
\begin{priklad}
\int \sin^3 x \cos^3 x \ud x
\end{priklad}
\res{$\frac{1}{3} \tg^3 x + C$}
\item
\begin{priklad}
\int \sin^2 x \cos^3 x \ud x
\end{priklad}
\res{$\frac{1}{3} \sin ^3 x - \frac{1}{5}\sin^5 x + C$}
\item
\begin{priklad}
\int \sin^4 x \ud x
\end{priklad}
\res{$\frac{3}{8} x - \frac{1}{4} \sin 2x + \frac{1}{32} \sin 4x + C$}
\item
\begin{priklad}
\int \sin 2x \cos 3x \ud x
\end{priklad}
\res{$\frac{1}{2} \cos x - \frac{1}{10}\cos 5x + C$}
\item
\begin{priklad}
\int \sin^2 x \sin 2x \ud x
\end{priklad}
\res{$\frac{1}{2} \sin^4 x + C$}
\item
\begin{priklad}
\int \cos 3x \cos 2x \ud x
\end{priklad}
\res{$\frac{1}{2} \sin x + \frac{1}{10} \sin 5x + C$}
\item
\begin{priklad}
\int \tg^2 3x \ud x
\end{priklad}
\res{$\frac{1}{3} \tg 3x - x + C$}
\item
\begin{priklad}
\int \frac{\ud x}{\cos^2 \pi x}
\end{priklad}
\res{$\frac{1}{\pi} \tg \pi x + C$}
\item
\begin{priklad}
\int \tg^3 x \ud x
\end{priklad}
\res{$\frac{1}{2} \tg^2 x + \ln |\cos x| + C$}
\item
\begin{priklad}
\int \frac{\tg^2 x}{\cos^2 x} \ud x
\end{priklad}
\res{$\frac{1}{3} \tg^3 x + C$}
\item
\begin{priklad}
\int \frac{1}{\sin^3 x } \ud x
\end{priklad}
\res{$- \frac{1}{2} \frac{1}{\sin x} \cotg x + \frac{1}{2} \ln \left | \frac{1}{\sin x} - \cotg x\right | + C$}
\item
\begin{priklad}
\int \frac{\cotg^3 x}{\sin^3 x} \ud x
\end{priklad}
\res{$- \frac{1}{5} \frac{1}{\sin^5x}+ \frac{1}{3} \frac{1}{ \sin^3 x} + C$}
\item
\begin{priklad}
\int \frac{\tg^4 x}{\cos^4 x} \ud x
\end{priklad}
\res{$\frac{1}{7} \tg^7 x + \frac{1}{5} \tg^5 x + C$}
\item
\begin{priklad}
\int \frac{\ud x}{(5-x^2)^{\frac32}}
\end{priklad}
\res{$\frac{x}{5\sqrt{5-x^2}} + C$}
\item
\begin{priklad}
\int \sqrt{x^2 -1} \ud x
\end{priklad}
\res{$\frac{1}{2} x \sqrt{x^2-1} - \frac{1}{2}\ln |x + \sqrt{x^2-1}| + C$}
\item
\begin{priklad}
\int \frac{x^2}{\sqrt{4-x^2}} \ud x
\end{priklad}
\res{$2 \arcsin \frac{x}{2} -\frac{1}{2}x \sqrt{4-x^2} + C$}
\item
\begin{priklad}
\int \frac{x}{(1-x^2) ^ {\frac32}} \ud x
\end{priklad}
\item
\begin{priklad}
\int \frac{x^2}{(1-x^2)^{\frac32}} \ud x
\end{priklad}
\res{$\frac{x}{\sqrt{1-x^2}} - \arcsin x + C$}
\item
\begin{priklad}
\int x \sqrt{4 - x^2} \ud x
\end{priklad}
\item
\begin{priklad}
\int \frac{e^x}{\sqrt{9-e^{2x}}} \ud x
\end{priklad}
\item
\begin{priklad}
\int \frac{\ud x}{x \sqrt{a^2 - x^2}}
\end{priklad}
\res{$\frac{1}{a} \ln \left | \frac{a-\sqrt{a^2 - x^2}}{x} \right | + C$}
\item
\begin{priklad}
\int e^x \sqrt{e^{2x} - 1} \ud x
\end{priklad}
\res{$\frac{1}{2} e^x \sqrt{e^{2x} -1} - \frac{1}{2} \ln(e^x + \sqrt{e^{2x} + 1}) + C$}
\item
\begin{priklad}
\int \frac{\ud x}{x^2 \sqrt{x^2-a^2}}
\end{priklad}
\res{$\frac{1}{a^2x} \sqrt{x^2-a^2}+C$}
\item
\begin{priklad}
\int \frac{\ud x}{e^x \sqrt{e^{2x} - 9}}
\end{priklad}
\res{$\frac{1}{9} e ^{-x}\sqrt{e^{2x} - 9} + C$}
\item
\begin{priklad}
\int \frac{\ud x}{(x^2 - 4x + 4)^{\frac32}}
\end{priklad}
\res{$- \frac{1}{2(x-2)^2} + C$}
\item
\begin{priklad}
\int \frac{x+3}{\sqrt{x^2+4x + 13}} \ud x
\end{priklad}
\res{$\sqrt{x^2+4x+13} + \ln(x+2+\sqrt{x^2+4x+13}) + C$}
\item
\begin{priklad}
\int x (8 - 2x - x^2) ^{\frac32} \ud x
\end{priklad}
\item
\begin{priklad}
\int \frac{\sqrt x}{ \sqrt{x} - 1} \ud x
\end{priklad}
\res{$x + 2 \sqrt{x} + 2 \ln |\sqrt{x} - 1| + C$}
\item
\begin{priklad}
\int \frac{\sqrt{x-1} + 1}{\sqrt{x-1} - 1} \ud x
\end{priklad}
\res{$x + 4 \sqrt{x-1}+ 4 \ln |\sqrt{x-1} - 1| + C$}
\item
\begin{priklad}
\int \frac{\ud x}{ 1 + e^{-x}}
\end{priklad}
\item
\begin{priklad}
\int 2x^2(4x + 1) ^ {-5/2} \ud x
\end{priklad}
\res{$\frac{1}{16}(4x+1) ^ {1/2} + \frac{1}{8}(4x+1) ^ {-1/2} - \frac{1}{48}(4x+1)^{-\frac32}+C$}
\item
\begin{priklad}
\int \ln ( x \sqrt{x}) \ud x
\end{priklad}
\res{$x \ln(x \sqrt{x}) - \frac{3}{2}x +C $}
\item
\begin{priklad}
\int \frac{x^3}{\sqrt{1+x^2}} \ud x
\end{priklad}
\res{$\frac{1}{3} (x^2-2) \sqrt{1+x^2}+ C$}
\item
\begin{priklad}
\int \frac{\sin 3x}{2 + \cos 3x} \ud x
\end{priklad}
\res{$- \frac{1}{3} \ln (2 + \cos 3x) + C$}
\item
\begin{priklad}
\int \frac{\sin x}{\cos^3 x} \ud x
\end{priklad}
\res{$\frac{1}{2} \tg^2 x + C$}
\item
\begin{priklad}
\int \frac{1- \sin 2x}{1+ \sin 2x} \ud x
\end{priklad}
\res{$\tg 2x - \frac{1}{\cos 2x} - x + C$}
\item
\begin{priklad}
\int \frac{\ud x}{\sqrt{x+1} - \sqrt{x}}
\end{priklad}
\res{$\frac{2}{3}(x+1) ^ {\frac32}+ \frac{2}{3} x ^ {\frac32} + C$}
\item
\begin{priklad}
\int x \ln \sqrt{x^2 + 1} \ud x
\end{priklad}
\item
\begin{priklad}
\int \frac{\sqrt{x^2 + 4}}{x} \ud x
\end{priklad}
\res{$2 \ln(\sqrt{x^2+4} - 2) - 2 \ln |x| + \sqrt{x^2+4} + C$}
\item
\begin{priklad}
\int x^2 \arcsin x \ud x
\end{priklad}
\res{$\frac{1}{3} x^3 \arcsin x + \frac{1}{3}(1-x^2)^{1/2} - \frac{1}{9}(1-x^2)^{\frac32} + C$}
\item
\begin{priklad}
\int \frac{\ud x}{ \sqrt{1- e ^{2x}}}
\end{priklad}
\res{$\ln |1 - \sqrt{1-e^{2x}}| - x + C$}
\item
\begin{priklad}
\int \frac{\sqrt{x}}{4(1+x^{\frac34})} \ud x
\end{priklad}
\res{$\frac13x^{\frac34}-\frac13 \ln(1+x^{\frac34})$}
\item
\begin{priklad}
\int \frac{\ud x}{1+\sqrt{x}}
\end{priklad}
\res{$-\ln(x-1)+2\sqrt{x}+ \ln(\sqrt{x}-1) -\ln(1+\sqrt{x})$}
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\odstavec{Zobecněný Riemannův integrál}
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\item Spočítejte
\begin{priklad}
\int_0^2 \frac{\ud x}{x^2-4x+3}
\end{priklad}
\res{Diverguje.}
\item Spočítejte
\begin{priklad}
\int_{-\infty}^{+\infty} \frac{\ud x}{(x^2+x+1)^2}
\end{priklad}
\res{$\frac49\pi\sqrt{3}$}
\item Spočítejte
\begin{priklad}
\int_2^{+\infty} \frac{\ud x}{x^2 + x -2}
\end{priklad}
\res{$\frac23\ln2$}
\item Spočítejte
\begin{priklad}
\int\limits_0^{+\infty} \frac{x^2-1}{1-x^4} \ud x
\end{priklad}
\res{$-\frac\pi2$}
\item Spočítejte
\begin{priklad}
\int_{-\infty}^{+\infty} \frac{\ud x}{(1+x^2)^2}
\end{priklad}
\res{$\frac\pi2$}
\item Spočítejte
\begin{priklad}
\int_{0}^{+\infty} \frac{\ud x}{(1+x^2)^3}
\end{priklad}
\res{$\frac{3\pi}{16}$}
\item Spočítejte
\begin{priklad}
\int_{0}^{+\infty} \frac{1-x^2}{(1-x^4)(1+x^2)}\ud x
\end{priklad}
\res{$\frac\pi4$}
\item Spočítejte
\begin{priklad}
\int_{1}^{+\infty} \frac{\ud~x}{x(1+x)}
\end{priklad}
\res{$\ln2$}
\item Spočítejte
\begin{priklad}
\int_{1}^{+\infty} \frac{\ud x}{x^2(1+x^2)}
\end{priklad}
\res{$1-\frac\pi4$}
\item Spočítejte
\begin{priklad}
\int_0^8 \frac{\ud x}{x^{2/3}}
\end{priklad}
\res{6}
\item Spočítejte
\begin{priklad}
\int_0^1 \frac{\ud x}{\sqrt{1-x^2}}
\end{priklad}
\res{$\frac{\pi}{2}$}
\item Spočítejte
\begin{priklad}
\int_0^2 \frac{x}{\sqrt{4-x^2}} \ud x
\end{priklad}
\res{2}
\item Spočítejte
\begin{priklad}
\int_3^5 \frac{x}{\sqrt{x^2-9}} \ud x
\end{priklad}
\res{4}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^{+\infty} e ^{px}\ud x
\end{priklad}
\res{Diverguje pro $p\geq0$. Jinak konverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_e^{+\infty} \frac{\ln x}{x} \ud x
\end{priklad}
\res{Diverguje.}
\item Spočítejte
\begin{priklad}
\int_0^1 x \ln x \ud x
\end{priklad}
\res{$ -\frac{1}{4}$}
\item Spočítejte
\begin{priklad}
\int_{-\infty}^{+\infty} \frac{\ud x}{1+x^2}
\end{priklad}
\res{$\pi$}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_{-\infty}^{+\infty} \frac{\ud x}{x^2}
\end{priklad}
\res{Diverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_{-3}^3 \frac{\ud x}{x(x+1)}
\end{priklad}
\res{Diverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_{-3}^1 \frac{\ud x}{x^2-4}
\end{priklad}
\res{Diverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^{+\infty} \cosh x \ud x
\end{priklad}
\res{Diverguje.}
\item Spočítejte
\begin{priklad}
\int_0^1 \ln x \ud x
\end{priklad}
\res{-1}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^2 \frac{x}{\sqrt{4-x^2}} \ud x
\end{priklad}
\res{2}
\item Spočítejte
\begin{priklad}
\int_0^{+\infty} \frac{\ud x}{1+x^3}
\end{priklad}
\res{$\frac{2\pi}{3\sqrt3}$}
\item Spočítejte
\begin{priklad}
\int_0^{+\infty} \frac{x^2+1}{x^4+1} \ud x
\end{priklad}
\res{$\frac{\pi}{\sqrt2}$}
\item Spočítejte
\begin{priklad}
\int_1^{+\infty} \frac{\sqrt{1+x^2}}{x^4} \ud x
\end{priklad}
\res{$\frac{2\sqrt2 - 1}{3}$}
\item Spočítejte
\begin{priklad}
\int_0^{+\infty} \frac{\arctg x}{(1+x^2)\sqrt{1+x^2}} \ud x
\end{priklad}
\res{$\frac{\pi}{2} - 1$}
\item Spočítejte
\begin{priklad}
\int_0^1 \frac{\arcsin x }{\sqrt{1-x^2}}
\end{priklad}
\res{$\frac{\pi^2}{8}$}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^{+\infty} \frac{x}{\sqrt{1+x^5}} \ud x
\end{priklad}
\res{Konverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_1^{+\infty} 2 ^{-x^2} \ud x
\end{priklad}
\res{Konverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^{+\infty}(1+x^5)^{-1/6} \ud x
\end{priklad}
\res{Diverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_{\pi}^{+\infty} \frac{\sin^2 2x}{x^2} \ud x
\end{priklad}
\res{Konverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_1^{+\infty} \frac{\ln x}{x^2} \ud x
\end{priklad}
\res{Konverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_e^{+\infty} \frac{\ud x}{\sqrt{x+1} \ln x}
\end{priklad}
\res{Diverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^{+\infty} \frac{x^2}{x^4-x^2+1}
\end{priklad}
\res{Konverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_1^{+\infty} \frac{\ud x}{x \sqrt[3]{x^2+1}}
\end{priklad}
\res{Konverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^{+\infty} \frac{x^2}{x^3+x^2+1} \ud x
\end{priklad}
\res{Diverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^{+\infty} \frac{\arctg x}{x^{\frac32}} \ud x
\end{priklad}
\res{Konverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^{+\infty} \frac{\ud x}{\sqrt{x} + x^2}
\end{priklad}
\res{Konverguje.}
\item Rozhodněte o konvergenci
\begin{priklad}
\int_0^{+\infty} \frac{\ln(1+x)}{x^{\frac32}} \ud x
\end{priklad}
\res{Konverguje.}
\item Spočítejte
\begin{priklad}
\int_0^{+\infty} \frac{1}{x\ln^2 x} \ud x
\end{priklad}
\res{Diverguje.}
\end{enumerate}