\documentclass{article}
\usepackage{cvika}

\begin{document}

\Nadpis{\bf 2. cvi�en� z MA --- 4. a 5.3.2009}

\nadpis{Primitivn� funkce alias neur�it� integr�ly}

Dopl�te n�sleduj�c� tabulku:

\def\vph{\vphantom{\vbox to 5mm{}}}

\begin{tabular*}{12cm}{c|c|c} 
 \hbox to 4cm{\hfill $f(x)$\hfill}& 
 \hbox to 4cm{\hfill $F(x)$\hfill}& 
 \hbox to 4cm{\hfill interval\hfill}\\
\hline
\vph $x^a$, $a \in \er \setminus \{-1\}$ && \\
\vph  $\frac 1x$ && \\
\vph  $e^x$ && \\
\vph  $\sin x$ && \\
\vph  $\cos x$ && \\
\vph  $\frac 1{\cos^2x}$ && \\
\vph  $\frac 1{\sqrt {1-x^2}}$ && \\
\vph  $\frac 1{1+x^2}$ && \\
\end{tabular*}

\medskip

Zopakujte si, jak se integruje substituc� a metodou per partes.
��m se li�� dv� v�ty o substituci?

P�i integrov�n� nezapome�te ur�it interval, na kter�m je
v�sledek platn�.


\pr (z�kladn�)
Tady by m�lo sta�it pou��t v��e uvedenou tabulku.

\cast $\int x^3+2x+\frac{17}x \d x$,
\cast $\int 18e^x+16e^{8x}-\frac1x+3\cos x \d x$,
\cast $\int\sqrt {x^6} \d x$,
\cast $\int{(1-x)^3\over x\root 3\of x}\d x$.

\pr (jednoduch� substituce)
Tady bude pot�eba n�jak� substituce -- kterou by m�lo b�t snadn� uh�dnout.

\cast $\int\root 3\of {1-3x}\d x$,
\cast $\int \sin^7x\cos x\d x$,
\cast $\int x e^{-x^2} \d x$,
\cast $\int \tg x\d x$,
\cast $\int {x^2\over(1-x)^{100}}\d x$,
\cast $\int \cotg x\d x$,
\cast $\int \frac{x}{1+x^4} \d x$,
\cast $\int \frac{x^2}{\cos x^3} \d x$,
\cast $\int \frac{1}{x \ln x} \d x$,
\cast $\int \frac{2x+1}{x^2 + x + 1} \d x$,
\cast $\int \sin^{2k+1} x \d x$,
\cast $\int \cos^{2k+1} x \d x$,
\cast $\int \cotg x \d x$,
\cast $\int \frac{1}{\sin x} \d x$,
\cast $\int \frac{1}{\cos x} \d x$.


\end{document}

