Součásti dokumentu Matematika1Priklady
Zdrojový kód
%\wikiskriptum{Matematika1Priklady}
\section{Neurčité integrály a primitivní funkce}
\subsection*{\fbox{Rozcvička}}
V této úvodní části jsou příklady na integrály, které pro svou nižší náročnost nebudou ve zkouškové písemce, a tudíž nejsou číslovány.
\begin{itemize}
\item \begin{priklad}
\int \sin (3x) \ud x
\end{priklad}
\item \begin{priklad}
\int \sqrt3 \sin x + \cos(2x) \ud x
\end{priklad}
\item \begin{priklad}
\int (4-\sqrt{x})^2 \ud x
\end{priklad}
\res{$16\,x+\frac{1}{2}\,{x}^{2}-\frac{16}{3}\,{x}^{3/2}+C$}
\item \begin{priklad}
\int x e^{-x^2}\ud x
\end{priklad}
\res{$-\frac{1}{2}\,{e^{-{x}^{2}}}+C$}
\end{itemize}
\subsection*{\fbox{Zkouškové příklady}}
\begin{enumerate}
\item
\begin{priklad}
\int x^{-\frac34}(x^\frac14 + 1) \ud x
\end{priklad}
\res{$2\,\sqrt {x}+4\,\sqrt [4]{x}+C$}
\item \begin{priklad}
\int x \cos(\pi x^2) \ud x
\end{priklad}
\res{$\frac {\sin \left( \pi \,{x}^{2} \right) }{2\pi } +C$}
\item \begin{priklad}
\int 3x^2(x^3+1)^\pi \ud x
\end{priklad}
\res{${\frac { \left( x+1 \right) \left( {x}^{2}-x+1 \right) \left( {x}^{3}+1 \right) ^{\pi }}{\pi +1}}+C = \frac{\left( x^3 + 1 \right) ^{\pi + 1}}{\pi + 1}+C$}
\item \begin{priklad}
\int \cos^4 x \sin x \ud x
\end{priklad}
\res{$-\frac{1}{5}\, \left( \cos{x} \right) ^{5}+C$}
\item \begin{priklad}
\int \sin^4 x \cos x \ud x
\end{priklad}
\res{$\frac{1}{5}\, \left( \sin{x} \right) ^{5}+C$}
\item \begin{priklad}
\int \frac{\cos\sqrt{x}}{\sqrt{x}} \ud x
\end{priklad}
\res{$2\,\sin \left( \sqrt {x} \right) +C$}
\item \begin{priklad}
\int \frac{x^2}{1+x^2}\ud x
\end{priklad}
\res{$x-\arctg{x} +C$}
\item \begin{priklad}
\int \frac{\sqrt{x} -2 \sqrt[3]{x^2} + 1}{\sqrt[4]{x}}\ud x
\end{priklad}
\res{$\frac45x^\frac54 - \frac{24}{17}x^\frac{17}{12} + \frac43x^\frac34 +C$}
\item \begin{priklad}
\int (2^x + 3^x) \ud x
\end{priklad}
\res{${\frac {{2}^{x}\ln \left( 3 \right) +{3}^{x}\ln \left( 2 \right) }{ \ln \left( 2 \right) \ln \left( 3 \right) }} + C$}
\item
\begin{priklad}
\int \max \{ 3, 2x^4\} \ud x
\end{priklad}
(Pozor na spojitost primitivní funkce!)
\res{TODO}
\item \begin{priklad}
\int \min \{ x^3, x\} \ud x
\end{priklad}
(Pozor na spojitost primitivní funkce!)
\res{TODO}
\item \begin{priklad}
\int \sqrt{1-\sin^2 x} \ud x
\end{priklad}
(Pozor na spojitost primitivní funkce!)
\res{TODO}
\item \begin{priklad}
\int \max\{1,x \} \ud x
\end{priklad}
(Pozor na spojitost primitivní funkce!)
\res{$x+C$, $\frac{x^2}{2}+D$, $C = -\frac12 +D$}
\item \begin{priklad}
\int \tgh^2 x \ud x
\end{priklad}
\res{$x-\tgh{x} +C$}
\item \begin{priklad}
\int \ctgh^2 x \ud x
\end{priklad}
\res{$x-\ctgh{x} +C$}
\item \begin{priklad}
\int \cotg^2 x \ud x
\end{priklad}
\res{$-\cotg{x} -x +C$}
\item \begin{priklad}
\int x (1-x^2)^6 \ud x
\end{priklad}
\res{$-\frac{1}{14}(1-x^2)^7+C$}
\item \begin{priklad}
\int \sin^5 x \cos x \ud x
\end{priklad}
\res{$\frac{1}{6}\, \left( \sin \left( x \right) \right) ^{6}+C$}
\item \begin{priklad}
\int \frac{\ud x}{e^x + e^{-x}}
\end{priklad}
\res{$\arctg\e^{x}+C$}
\item \begin{priklad}
\int \frac{\ud x}{x (\ln x + 3)}
\end{priklad}
\res{$\ln \left( \ln \left( x \right) +3 \right) +C$}
\item \begin{priklad}
\int \frac{\arctg x}{x^2+1} \ud x
\end{priklad}
\res{$\frac{1}{2}\, \left( \arctg \left( x \right) \right) ^{2}+C$}
\item \begin{priklad}
\int \cos^4 x \ud x
\end{priklad}
\res{$\frac{1}{4}\, \left( \cos \left( x \right) \right) ^{3}\sin \left( x \right)
+\frac{3}{8}\,\cos \left( x \right) \sin \left( x \right) +\frac{3}{8}\,x
+C$}
\item \begin{priklad}
\int \cos^3 x \ud x
\end{priklad}
\res{$\frac{1}{3}\, \left( \cos \left( x \right) \right) ^{2}\sin \left( x \right)
+\frac{2}{3}\,\sin \left( x \right) +C$}
\item \begin{priklad}
\int \sin^6 x \ud x
\end{priklad}
\res{$-\frac{1}{6}\, \left( \sin \left( x \right) \right) ^{5}\cos \left( x
\right) -{\frac {5}{24}}\, \left( \sin \left( x \right) \right) ^{3}
\cos \left( x \right) -{\frac {5}{16}}\,\cos \left( x \right) \sin
\left( x \right) +{\frac {5}{16}}\,x+C$}
\item \begin{priklad}
\int \sqrt{x} \ln x \ud x
\end{priklad}
\res{$\frac{2}{3}\,{x}^{3/2}\ln \left( x \right) -\frac{4}{9}\,{x}^{3/2}+C$}
\item \begin{priklad}
\int \arctg x \ud x
\end{priklad}
\res{$x~\arctg{x} -\frac{1}{2}\,\ln \left( 1+{x}^{2} \right) +C$}
\item \begin{priklad}
\int x~\arctg x \ud x
\end{priklad}
\res{$\frac{1}{2}\,{x}^{2}\arctg{x} -\frac{1}{2}\,x+ \frac{1}{2}\,\arctg{x} +C$}
%\item \begin{priklad}
%\int \frac{xe^x}{(x+1)^2}\ud x
%\end{priklad}
%\res{${\frac {{e^{x}}}{x+1}}+C$}
\item \begin{priklad}
\int \frac{\ud x}{\sqrt{1-3x^2}}
\end{priklad}
\res{$\frac{\sqrt{3}}{3}\arcsin \left( \sqrt {3}x \right) +C$}
\item \begin{priklad}
\int \frac{\ud x}{\sqrt{7+x-x^2}}
\end{priklad}
\res{$\arcsin \left( {\frac {2\sqrt {29}}{29}} \left( x-\frac{1}{2} \right) \right) +C$}
\item \begin{priklad}
\int \frac{5x+1}{\sqrt{3-x^2}}\ud x
\end{priklad}
\res{$-5\,\sqrt {3-{x}^{2}}+\arcsin \left( \frac{\sqrt {3}}{3}x \right) +C$}
\item \begin{priklad}
\int \sqrt{5+x-x^2} \ud x
\end{priklad}
\res{$-\frac{1}{4}\, \left( 1-2\,x \right) \sqrt {5+x-{x}^{2}}+{\frac {21}{8}}\,\arcsin \left( \frac{2\sqrt {21}}{21} \left( x-\frac{1}{2} \right) \right) +C$}
\item \begin{priklad}
\int \frac{x^2+5x}{x^2-1}\ud x
\end{priklad}
\res{$x+3\,\ln \left( x-1 \right) +2\,\ln \left( x+1 \right) +C$}
\item \begin{priklad}
\int \tg x \ud x
\end{priklad}
\res{$-\ln \left( \cos \left( x \right) \right) +C$}
\item \begin{priklad}
\int \frac{\ln^2 x}{x^2}\ud x
\end{priklad}
\res{$-{\frac { \left( \ln \left( x \right) \right) ^{2}}{x}}-2\,{\frac {\ln \left( x \right) }{x}}-2\,{x}^{-1} +C$}
\item \begin{priklad}
\int \frac{x}{\sqrt{1-x^2}}\ud x
\end{priklad}
\res{$-\sqrt {1-{x}^{2}}+C$}
\item \begin{priklad}
\int \cos^5 x\sqrt{\sin x}\ud x
\end{priklad}
\res{${\frac {2}{231}}\, \left( \sin \left( x \right) \right) ^{3/2}
\left( 32+21\, \left( \cos \left( x \right) \right) ^{4}+24\,
\left( \cos \left( x \right) \right) ^{2} \right) +C$}
\item \begin{priklad}
\int \frac{\cos 3x}{2+ \sin 3x} \ud x
\end{priklad}
\res{$\frac{1}{3}\,\ln \left( 2+\sin \left( 3\,x \right) \right) +C$}
\item
Nalezněte $f(x)$, znáte-li:
$f''(x) = \cos{x}$,
$f'(0) = 1$,
$f(0) = 2$.
\res{$f(x) = x-\cos x +3$}
\item \begin{priklad}
\int \frac{(1-x)^3}{x\sqrt[3]{x}}\ud x
\end{priklad}
\res{$3x^{-1/3}(-1-\frac{3}{2}x+\frac{3}{5}x^2+\frac{1}{8}x^3)+C$}
\item \begin{priklad}
\int \frac{e^{3x} + 1}{e^x + 1}\ud x
\end{priklad}
\res{$\frac{e^{2x}}{2}-e^x+x+C$}
\item \begin{priklad}
\int (x + |x|)^2 \ud x
\end{priklad}
\res{$\frac{2}{3}(x^3 + |x^3|)+C$}
\item \begin{priklad}
\int \frac{x}{(x^2-1)^{3/2}}\ud x
\end{priklad}
\res{$-\frac{1}{\sqrt{x^2-1}}+C$}
\item \begin{priklad}
\int \frac{x}{4+x^4}\ud x
\end{priklad}
\res{$\frac{1}{4} \arctg \frac{x^2}{2}+C$}
\item \begin{priklad}
\int \frac{\sin x}{\sqrt{\cos^3 x}}\ud x
\end{priklad}
\res{$\frac{2}{\sqrt{\cos x}}+C$}
\item \begin{priklad}
\int x e^{-x^2}\ud x
\end{priklad}
\res{$-\frac{1}{2}\,e ^{-x^2}+C$}
\item \begin{priklad}
\int \frac{\ln^2 x}{x}\ud x
\end{priklad}
\res{$ \frac{\ln^3 x}{3}+C$}
\item \begin{priklad}
\int \frac{\ln x}{x\sqrt{1+\ln x}} \ud x
\end{priklad}
\res{$\frac{2}{3}\sqrt{1+\ln x}(\ln x -2)+C$}
\item \begin{priklad}
\int \frac{1}{\sin^2 x (1+\tg x)} \ud x
\end{priklad}
\res{$\ln |1+\cot x| - \cotg x+C$}
\item \begin{priklad}
\int \frac{\sin x \cos^3 x}{1+\cos^2 x} \ud x
\end{priklad}
\res{$-\frac{1}{2}\cos^2x + \frac{1}{2} \ln(1+\cos^2x) +C$}
\item \begin{priklad}
\int \sqrt{x} \ln^2 x \ud x
\end{priklad}
\res{$\frac{2}{27}x^{3/2}(9 \ln^2x - 12 \ln x +8)+C$}
\item \begin{priklad}
\int x \sinh x \ud x
\end{priklad}
\res{$x \cosh x - \sinh x+C$}
\item \begin{priklad}
\int x^2 \arccos x \ud x
\end{priklad}
\res{$\frac{1}{3} x ^3 \arccos x + \frac{1}{9}(1-x^2)^{3/2} - \frac{1}{3}(1-x^2)^{1/2}+C$}
\item \begin{priklad}
\int \arctg \sqrt x \ud x
\end{priklad}
\res{$x \arctg \sqrt x + \arctg \sqrt x -\sqrt x+C$}
\item \begin{priklad}
\int \frac{\ln \sin x}{\sin^2 x} \ud x
\end{priklad}
\res{$-\cotg{ x} \ln \sin x - \cotg x -x+C$}
\item \begin{priklad}
\int x e^{-x} \ud x
\end{priklad}
\res{$-x e^{-x} - e^{-x}+C$}
\item \begin{priklad}
\int x^2 e^{-x} \ud x
\end{priklad}
\res{$-e^{-x}(x^2+2x+2)+C$}
\item \begin{priklad}
\int \frac{x^2}{\sqrt{1-x}} \ud x
\end{priklad}
\res{$-2x^2(1-x)^{1/2} - \frac{8}{3}x(1-x)^{3/2}-\frac{16}{15}(1-x)^{5/2}+C$}
\item \begin{priklad}
\int x \ln \sqrt x \ud x
\end{priklad}
\res{$ \frac{1}{4} x^2 \ln x - \frac{1}{8}x^2+C$}
\item \begin{priklad}
\int \frac{\ln(x+1)}{\sqrt{x+1}} \ud x
\end{priklad}
\res{$ 2\sqrt{x+1} \ln (x+1) -4\sqrt{x+1}+C$}
\item \begin{priklad}
\int \ln^2 x \ud x
\end{priklad}
\res{$x \ln^2 x - 2x \ln x + 2x +C$}
\item \begin{priklad}
\int x^3 3 ^x \ud x
\end{priklad}
\res{$3^x (\frac{x^3}{\ln 3} - \frac{3x^2}{\ln^2 3} + \frac{6x}{\ln^3 3} - \frac{6}{\ln^4 3})+C$}
\item \begin{priklad}
\int x^3 \sin x^2 \ud x
\end{priklad}
\res{$-\frac{1}{2}x^2 \cos x^2 + \frac{1}{2}\sin x^2 + C$}
\item \begin{priklad}
\int \ln (1+x^2) \ud x
\end{priklad}
\res{$x \ln (1+x^2) - 2x +2 \arctg x+C$}
\item \begin{priklad}
\int \cotg(\pi -x) \ud x
\end{priklad}
\res{$-\ln \lvert \sin (x) \rvert + C$}
\item \begin{priklad}
\int \cot x \ln \sin x \ud x
\end{priklad}
\res{$\frac{1}{2}(\ln \sin x)^2+C$}
\item \begin{priklad}
\int \frac{1}{\cos^2 x(9+\tg^2 x)}\ud x
\end{priklad}
\res{$\frac13 \arctg(\frac{1}{3} \tg x) +C$}
\item \begin{priklad}
\int \frac{1}{\cos^2 x\sqrt{9-\tg^2 x}}\ud x
\end{priklad}
\res{$\arcsin \left( \frac{1}{3}\, \tg{x} \right) +C$}
\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)^{3/2}}\ud x
\end{priklad}
\res{$\frac{1}{\sqrt{1-x^2}} +C$}
\item \begin{priklad}
\int x\sqrt{4-x^2}\ud x
\end{priklad}
\res{$-\frac{1}{3}(4-x^2)^{3/2} +C$}
\item \begin{priklad}
\int \frac{\ud x}{x\sqrt{a^2-x^2}}
\end{priklad}
\res{$\frac{1}{a} \ln \Big|\frac{a-\sqrt{a^2-x^2}}{x}\Big| +C$}
\item \begin{priklad}
\int \frac{\ud x}{x^2\sqrt{a^2+x^2}}
\end{priklad}
\res{$-\frac{1}{a^2x}\sqrt{a^2+x^2} +C$}
\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$}
\item \begin{priklad}
\int x \sqrt{6x-x^2-8} \ud x
\end{priklad}
\res{$-\frac{1}{3}(6x-x^2-8)^{3/2}+\frac{3}{2}\arcsin(x-3) + \frac{3}{2}\sqrt{6x-x^2-8} +C$}
\item \begin{priklad}
\int \frac{x}{(x^2+2x+5)^2}\ud x
\end{priklad}
\res{$\frac{x^2+x}{8(x^2+2x+5)} - \frac{1}{16}\arctg\big( \frac{x+1}{2} \big) +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$}
\item \begin{priklad}
\int \sqrt{6x-x^2-8} \ud x
\end{priklad}
\res{$\frac{1}{2}(x-3)\sqrt{6x-x^2-8} + \frac{1}{2} \arcsin(x-3) +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)^{3/2} +C$}
\item \begin{priklad}
\int \frac{3}{\sqrt{2-3x-4x^2}} \ud x
\end{priklad}
\res{$\frac{3}{2} \arcsin \big( \frac{8x+3}{\sqrt{41}} \big) +C$}
\item \begin{priklad}
\int \frac{x^2}{\sqrt{3-2x-x^2}} \ud x
\end{priklad}
\res{$-\frac{1}{2}\,x\sqrt {3-2\,x-{x}^{2}}+\frac{3}{2}\,\sqrt {3-2\,x-{x}^{2}}+3\,\arcsin \left( \frac{x+1}{x} \right) +C$}
\item \begin{priklad}
\int \frac{\hbox{arctg}(\ln x)}{x} \ud x
\end{priklad}
\res{$\ln x~ \arctg \ln x - \frac{1}{2} \ln(\ln^2x+1) + C$}
\item \begin{priklad}
\int \sqrt{\frac{1+x^2}{\left(1-x^4\right) \arcsin x}} \ud x
\end{priklad}
\res{$2\,\sqrt{\arcsin x} +C$}
\item \begin{priklad}
\int \arctg \sqrt{x^2 - 1} \ud x
\end{priklad}
\res{$x\,\arctg \sqrt{x^2-1} - \argcosh x +C$}
\item \begin{priklad}
\int \cos{x}\sin^5(x) \ud x
\end{priklad}
\res{$\frac{1}{6}\sin^6(x) + C$}
\item \begin{priklad}
\int \frac{\tg x + \cotg x}{\sin(2x)} \ud x
\end{priklad}
\res{$-\cotg(2x) + C$}
\item Nalezněte všechny funkce, které mají tu vlastnost, že $\displaystyle f''(x) = e^x + 1$.
\res{$f(x) = e^x + Cx +D + \frac{1}{2} x^2$}
\item $\displaystyle \int x2^{x^2+1} \ud x$
\res{$2^{x^2}/\ln2+C$}
\item Nalezněte primitivní funkci k funkci $\displaystyle f(x) = \frac{x}{4+x^4}$.
\res{$\frac{1}{4}\,\arctg(\frac{1}{2} x^2)+C$}
\item Nalezněte všechny funkce $f$, které mají tu vlastnost, že $\displaystyle f''(x) = e^x + \frac{1}{x^2}$.
\res{ $f(x) = \e^x + Cx + D - \ln x$}
\item Nalezněte $f(x)$, znáte-li
\begin{priklad}
f^\prime(x) = 2x - 1; f(3) = 4\,.
\end{priklad}
\res{$x^2 - x - 2$}
\item Nalezněte $f(x)$, znáte-li
\begin{priklad}
f^{\prime\prime}(x) = \cos {x}; f^\prime(0) = 1; f(0) = 2\,.
\end{priklad}
\res{$x - \cos{x} + 3$}
\item Nalezněte $f(x)$, znáte-li
\begin{priklad}
f^{\prime\prime}(x) = 6x - 2; f^\prime(0) = 1; f(0) = 2\,.
\end{priklad}
\res{$x^3 - x^2 + x + 2$}
\item Nalezněte $f(x)$, znáte-li
\begin{priklad}
f^{\prime\prime}(x) = 2x - 3; f(2) = -1; f(0) = 3\,.
\end{priklad}
\res{$\frac{x^3}{3} - \frac{3x^2}{2} - \frac{x}{3} + 3$}
\item \begin{priklad}
\int \frac{t}{(4t^2+9)^2} \ud t
\end{priklad}
\res{$-\frac{1}{8(4t^2 + 9)}+C$}
\item \begin{priklad}
\int x^{-\frac12} \sin (x^{\frac12}) \ud x
\end{priklad}
\res{$-2 \cos(x^{1/2})+C$}
\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 \frac{\sqrt x}{1+x \sqrt{x} } \ud x
\end{priklad}
\res{$\frac{2}{3}\ln|1+x\sqrt x|+C$}
\item \begin{priklad}
\int \frac{e^{\frac1x}}{x^2} \ud x
\end{priklad}
\res{$ - e ^{\frac1x}+C$}
\item \begin{priklad}
\int \frac{\log_2{x^3}}{x} \ud x
\end{priklad}
\res{$\frac{3}{\ln 4}(\ln x)^2+C$}
\item \begin{priklad}
\int \frac{\cos^2 x^2 + \ln^2 x^2 + \sin^2 x^2}{x} \ud x
\end{priklad}
\res{$\ln |x| + \frac{4}{3}\ln^3 |x| +C$}
\end{enumerate}
