Matematika1Priklady:Kapitola5: Porovnání verzí
Z WikiSkripta FJFI ČVUT v Praze
Řádka 1: | Řádka 1: | ||
%\wikiskriptum{Matematika1Priklady} | %\wikiskriptum{Matematika1Priklady} | ||
− | \section{ | + | \section{Neurčité integrály a primitivní funkce} |
− | \begin{ | + | |
+ | \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{$16\,x+1/2\,{x}^{2}-16/3\,{x}^{3/2}+C$} | ||
+ | |||
+ | \item \begin{priklad} | ||
+ | \int x e^{-x^2}\ud x | ||
+ | \end{priklad} | ||
+ | \res{$-1/2\,{e^{-{x}^{2}}}+C$} | ||
+ | |||
+ | \end{itemize} | ||
+ | |||
+ | \subsection*{\fbox{Zkouškové příklady}} | ||
+ | |||
\begin{enumerate} | \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{$1/2\,{\frac {\sin \left( \pi \,{x}^{2} \right) }{\pi }} +C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int 3x^2(x^3+1)^\pi \ud x | |
− | + | \end{priklad} | |
− | + | \res{${\frac { \left( x+1 \right) \left( {x}^{2}-x+1 \right) \left( {x}^{3}+1 \right) ^{\pi }}{\pi +1}}+C$} | |
− | + | ||
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \cos^4 x \sin x \ud x | |
− | + | \end{priklad} | |
− | + | \res{$-1/5\, \left( \cos{x} \right) ^{5}+C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \sin^4 x \cos x \ud x | |
− | + | \end{priklad} | |
− | + | \res{$1/5\, \left( \sin{x} \right) ^{5}+C$} | |
− | + | ||
− | + | ||
− | + | \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 $\displaystyle \int \max\{1,x \} \ud x$. (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{$-\cotg \left( x \right) +1/2\,\pi -x +C$} | |
− | + | ||
− | + | \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{$1/6\, \left( \sin \left( x \right) \right) ^{6}+C$} | |
− | + | ||
− | + | \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{$1/2\, \left( \arctg \left( x \right) \right) ^{2}+C$} | |
− | + | ||
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \cos^4 x \ud x | |
− | + | \end{priklad} | |
− | + | \res{$1/4\, \left( \cos \left( x \right) \right) ^{3}\sin \left( x \right) | |
− | + | +3/8\,\cos \left( x \right) \sin \left( x \right) +3/8\,x | |
− | + | +C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \cos^3 x \ud x | |
− | + | \end{priklad} | |
− | + | \res{$1/3\, \left( \cos \left( x \right) \right) ^{2}\sin \left( x \right) | |
− | + | +2/3\,\sin \left( x \right) +C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \sin^6 x \ud x | |
− | + | \end{priklad} | |
− | + | \res{$-1/6\, \left( \sin \left( x \right) \right) ^{5}\cos \left( x | |
− | + | \right) -{\frac {5}{24}}\, \left( \sin \left( x \right) \right) ^{3} | |
− | + | \cos \left( x \right) -{\frac {5}{16}}\,\cos \left( x \right) \sin | |
− | + | \left( x \right) +{\frac {5}{16}}\,x+C$} | |
− | + | ||
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \sqrt{x} \ln x \ud x | |
− | + | \end{priklad} | |
− | + | \res{$2/3\,{x}^{3/2}\ln \left( x \right) -4/9\,{x}^{3/2}+C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \arctg x \ud x | |
− | + | \end{priklad} | |
− | + | \res{$\arctg \left( x \right) x-1/2\,\ln \left( 1+{x}^{2} \right) +C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int x \arctg x \ud x | |
− | + | \end{priklad} | |
− | + | \res{$1/2\,\arctg \left( x \right) {x}^{2}-1/2\,x+1/2\,\arctg \left( x \right) +C$} | |
− | + | ||
− | + | \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{$1/3\,\sqrt {3}\arcsin \left( \sqrt {3}x \right) +C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \frac{\ud x}{\sqrt{7+x-x^2}} | |
− | + | \end{priklad} | |
− | + | \res{$\arcsin \left( {\frac {2}{29}}\,\sqrt {29} \left( x-1/2 \right) \right) +C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \frac{5x+1}{\sqrt{3-x^2}}\ud x | |
− | + | \end{priklad} | |
− | + | \res{$-5\,\sqrt {3-{x}^{2}}+\arcsin \left( 1/3\,\sqrt {3}x \right) +C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \sqrt{5+x-x^2} \ud x | |
− | + | \end{priklad} | |
− | + | \res{$-1/4\, \left( 1-2\,x \right) \sqrt {5+x-{x}^{2}}+{\frac {21}{8}}\,\arcsin \left( 2/21\,\sqrt {21} \left( x-1/2 \right) \right) +C$} | |
− | + | ||
− | + | \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{$1/3\,\ln \left( 2+\sin \left( 3\,x \right) \right) +C$} | |
− | + | ||
− | + | \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{$3x^{-1/3}(-1-3/2x+3/5x^2+1/8x^3)+C$} | |
− | + | ||
− | + | \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{$-1/2 e ^{-x^2}+C$} | |
− | + | ||
− | + | \item \begin{priklad} | |
− | + | \int \frac{\ln^2 x}{x}\ud x | |
− | + | \end{priklad} | |
+ | \res{$ \frac{\ln^3 x}{3}+C$} | ||
+ | |||
+ | \item \begin{priklad} | ||
+ | \int \frac{\ln x}{x\sqrt{1+\ln x}} | ||
+ | \end{priklad} | ||
+ | \res{$\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}{9}(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$} | ||
+ | |||
+ | \item \begin{priklad} | ||
+ | \int \frac{\ln \sin x}{\sin^2 x} | ||
+ | \end{priklad} | ||
+ | \res{$-\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$} | ||
+ | |||
+ | \item \begin{priklad} | ||
+ | \int \frac{x^2}{\sqrt{1-x}} | ||
+ | \end{priklad} | ||
+ | \res{$-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{$-\ln \left( \sin \left( x \right) \right) +C$} | ||
+ | |||
+ | \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{$\arctg(\frac{1}{3} \tg x) +C$} | ||
+ | |||
+ | \item \begin{priklad} | ||
+ | \int \frac{1}{\cos^2 x\sqrt{9-\tg^2 x}}\ud x | ||
+ | \end{priklad} | ||
+ | \res{$\arcsin \left( 1/3\, \tg{x} \right) +C$} | ||
+ | |||
+ | \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$} | ||
+ | |||
+ | \item \begin{priklad} | ||
+ | \int \frac{x+3}{\sqrt{x^2+4x+13}} | ||
+ | \end{priklad} | ||
+ | \res{$\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$} | ||
+ | |||
+ | \item \begin{priklad} | ||
+ | \int \frac{x^2}{\sqrt{3-2x-x^2}}\ud x | ||
+ | \end{priklad} | ||
+ | \res{$-1/2\,x\sqrt {3-2\,x-{x}^{2}}+3/2\,\sqrt {3-2\,x-{x}^{2}}+3\,\arcsin \left( 1/2\,x+1/2 \right) +C$} | ||
+ | |||
+ | \item | ||
+ | $\displaystyle \int \frac{\hbox{arctg}(\ln x)}{x} \ud x$ | ||
+ | \res{$\ln x~ \arctg \ln x - 1/2 \ln(\ln^2x+1) + C$} | ||
+ | |||
+ | \item $\displaystyle \int \ln^2 x \ud x$ | ||
+ | \res{$x \ln^2 x - 2x\ln x + 2x + C$ } | ||
+ | |||
+ | \item $\displaystyle \int x~ \arctg x~\ud x$. | ||
+ | \res{$1/2 x^2 \arctg x-1/2x+1/2\arctg x+C$} | ||
+ | |||
+ | \item $\displaystyle \int \cos{x}\sin^5(x)~\ud x$. | ||
+ | \res{$\sin^6(x)/6 + C$} | ||
+ | |||
+ | \item $\displaystyle \int \arctg x \ud x$. | ||
+ | \res{$x~\arctg x - 1/2 \ln(1+x^2)+C$} | ||
+ | |||
+ | \item Nalezněte všechny funkce, které mají tu vlastnost, že $\displaystyle f''(x) = e^x + 1$. | ||
+ | |||
+ | \res{$f(x) = e^x + Cx +D + 1/2 x^2$} | ||
+ | |||
+ | |||
+ | |||
+ | \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{$1/4 \arctg(1/2 x^2)+C$} | ||
+ | |||
+ | |||
+ | \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) = bx - 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} dt | ||
+ | \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$} | ||
+ | |||
\end{enumerate} | \end{enumerate} | ||
− | |||
− | |||
− | |||
− | |||
− |
Verze z 10. 7. 2011, 12:47
[ znovu generovat, | výstup z překladu ] | Kompletní WikiSkriptum včetně všech podkapitol. | |
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Součásti dokumentu Matematika1Priklady
součást | akce | popis | poslední editace | soubor | |||
---|---|---|---|---|---|---|---|
Hlavní dokument | editovat | Hlavní stránka dokumentu Matematika1Priklady | Fucikrad | 18. 9. 2011 | 08:54 | ||
Řídící stránka | editovat | Definiční stránka dokumentu a vložených obrázků | Admin | 7. 9. 2015 | 14:44 | ||
Header | editovat | Hlavičkový soubor | Fucikrad | 27. 4. 2022 | 09:11 | header.tex | |
Kapitola1 | editovat | Limity a spojitost | Pitrazby | 25. 10. 2016 | 09:25 | kapitola1.tex | |
Kapitola2 | editovat | Derivace, inverzní funkce, tečny, normály, asymptoty | Dvoraro3 | 4. 11. 2022 | 22:56 | kapitola2.tex | |
Kapitola3 | editovat | Vyšetřování funkcí | Admin | 29. 1. 2023 | 20:44 | kapitola3.tex | |
Kapitola4 | editovat | Extremální úlohy, konvexnost, konkávnost, inflexe | Admin | 3. 4. 2024 | 11:17 | kapitola4.tex | |
Kapitola5 | editovat | Neurčité integrály a primitivní funkce | Dvoraro3 | 28. 11. 2022 | 23:16 | kapitola5.tex | |
Kapitola6 | editovat | Určité integrály | Pitrazby | 28. 4. 2016 | 12:29 | kapitola6.tex | |
Kapitola7 | editovat | Aplikace integrálů | Fucikrad | 12. 4. 2022 | 10:53 | kapitola7.tex |
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{$16\,x+1/2\,{x}^{2}-16/3\,{x}^{3/2}+C$} \item \begin{priklad} \int x e^{-x^2}\ud x \end{priklad} \res{$-1/2\,{e^{-{x}^{2}}}+C$} \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{$1/2\,{\frac {\sin \left( \pi \,{x}^{2} \right) }{\pi }} +C$} \item \begin{priklad} \int 3x^2(x^3+1)^\pi \ud x \end{priklad} \res{${\frac { \left( x+1 \right) \left( {x}^{2}-x+1 \right) \left( {x}^{3}+1 \right) ^{\pi }}{\pi +1}}+C$} \item \begin{priklad} \int \cos^4 x \sin x \ud x \end{priklad} \res{$-1/5\, \left( \cos{x} \right) ^{5}+C$} \item \begin{priklad} \int \sin^4 x \cos x \ud x \end{priklad} \res{$1/5\, \left( \sin{x} \right) ^{5}+C$} \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 $\displaystyle \int \max\{1,x \} \ud x$. (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{$-\cotg \left( x \right) +1/2\,\pi -x +C$} \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{$1/6\, \left( \sin \left( x \right) \right) ^{6}+C$} \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{$1/2\, \left( \arctg \left( x \right) \right) ^{2}+C$} \item \begin{priklad} \int \cos^4 x \ud x \end{priklad} \res{$1/4\, \left( \cos \left( x \right) \right) ^{3}\sin \left( x \right) +3/8\,\cos \left( x \right) \sin \left( x \right) +3/8\,x +C$} \item \begin{priklad} \int \cos^3 x \ud x \end{priklad} \res{$1/3\, \left( \cos \left( x \right) \right) ^{2}\sin \left( x \right) +2/3\,\sin \left( x \right) +C$} \item \begin{priklad} \int \sin^6 x \ud x \end{priklad} \res{$-1/6\, \left( \sin \left( x \right) \right) ^{5}\cos \left( x \right) -{\frac {5}{24}}\, \left( \sin \left( x \right) \right) ^{3} \cos \left( x \right) -{\frac {5}{16}}\,\cos \left( x \right) \sin \left( x \right) +{\frac {5}{16}}\,x+C$} \item \begin{priklad} \int \sqrt{x} \ln x \ud x \end{priklad} \res{$2/3\,{x}^{3/2}\ln \left( x \right) -4/9\,{x}^{3/2}+C$} \item \begin{priklad} \int \arctg x \ud x \end{priklad} \res{$\arctg \left( x \right) x-1/2\,\ln \left( 1+{x}^{2} \right) +C$} \item \begin{priklad} \int x \arctg x \ud x \end{priklad} \res{$1/2\,\arctg \left( x \right) {x}^{2}-1/2\,x+1/2\,\arctg \left( x \right) +C$} \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{$1/3\,\sqrt {3}\arcsin \left( \sqrt {3}x \right) +C$} \item \begin{priklad} \int \frac{\ud x}{\sqrt{7+x-x^2}} \end{priklad} \res{$\arcsin \left( {\frac {2}{29}}\,\sqrt {29} \left( x-1/2 \right) \right) +C$} \item \begin{priklad} \int \frac{5x+1}{\sqrt{3-x^2}}\ud x \end{priklad} \res{$-5\,\sqrt {3-{x}^{2}}+\arcsin \left( 1/3\,\sqrt {3}x \right) +C$} \item \begin{priklad} \int \sqrt{5+x-x^2} \ud x \end{priklad} \res{$-1/4\, \left( 1-2\,x \right) \sqrt {5+x-{x}^{2}}+{\frac {21}{8}}\,\arcsin \left( 2/21\,\sqrt {21} \left( x-1/2 \right) \right) +C$} \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{$1/3\,\ln \left( 2+\sin \left( 3\,x \right) \right) +C$} \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{$3x^{-1/3}(-1-3/2x+3/5x^2+1/8x^3)+C$} \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{$-1/2 e ^{-x^2}+C$} \item \begin{priklad} \int \frac{\ln^2 x}{x}\ud x \end{priklad} \res{$ \frac{\ln^3 x}{3}+C$} \item \begin{priklad} \int \frac{\ln x}{x\sqrt{1+\ln x}} \end{priklad} \res{$\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}{9}(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$} \item \begin{priklad} \int \frac{\ln \sin x}{\sin^2 x} \end{priklad} \res{$-\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$} \item \begin{priklad} \int \frac{x^2}{\sqrt{1-x}} \end{priklad} \res{$-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{$-\ln \left( \sin \left( x \right) \right) +C$} \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{$\arctg(\frac{1}{3} \tg x) +C$} \item \begin{priklad} \int \frac{1}{\cos^2 x\sqrt{9-\tg^2 x}}\ud x \end{priklad} \res{$\arcsin \left( 1/3\, \tg{x} \right) +C$} \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$} \item \begin{priklad} \int \frac{x+3}{\sqrt{x^2+4x+13}} \end{priklad} \res{$\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$} \item \begin{priklad} \int \frac{x^2}{\sqrt{3-2x-x^2}}\ud x \end{priklad} \res{$-1/2\,x\sqrt {3-2\,x-{x}^{2}}+3/2\,\sqrt {3-2\,x-{x}^{2}}+3\,\arcsin \left( 1/2\,x+1/2 \right) +C$} \item $\displaystyle \int \frac{\hbox{arctg}(\ln x)}{x} \ud x$ \res{$\ln x~ \arctg \ln x - 1/2 \ln(\ln^2x+1) + C$} \item $\displaystyle \int \ln^2 x \ud x$ \res{$x \ln^2 x - 2x\ln x + 2x + C$ } \item $\displaystyle \int x~ \arctg x~\ud x$. \res{$1/2 x^2 \arctg x-1/2x+1/2\arctg x+C$} \item $\displaystyle \int \cos{x}\sin^5(x)~\ud x$. \res{$\sin^6(x)/6 + C$} \item $\displaystyle \int \arctg x \ud x$. \res{$x~\arctg x - 1/2 \ln(1+x^2)+C$} \item Nalezněte všechny funkce, které mají tu vlastnost, že $\displaystyle f''(x) = e^x + 1$. \res{$f(x) = e^x + Cx +D + 1/2 x^2$} \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{$1/4 \arctg(1/2 x^2)+C$} \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) = bx - 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} dt \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$} \end{enumerate}