\documentclass{article}
\usepackage{cvika}

\begin{document}

\Nadpis{\bf 3. cvi�en� z MA --- 11. a 12.3.2009}

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

\pr (per partes)

\cast $\int x^3 \ln x \d x$,
\cast $\int \sin^n x \d x$ (rekurentn� formul�),
\cast $\int \cos^n x \d x$ (rekurentn� formul�),
\cast $\int e^x x^n  \d x$ (rekurentn� formul�),
\cast $\int e^x \sin x \d x$,
\cast $\int x \sin x \d x$,
\cast $\int x^a \ln x \d x$ (kde $a > 0$),
\cast $\int \arcsin x \d x$, 
\cast $\int \arctg x \d x$, 
\cast $\int x^2/e^x \d x$, 
\cast $\int \sqrt {1-x^2} \d x$. 

\pr (trochu t쾹�)

\cast $\int \tg^2 x \d x$.
\cast $\int \cotg^2 x \d x$.
\cast $\int \ln (x + \sqrt {1+x^2}) \d x$.

\pr Vy�et�ete, jak� vlastnosti mus� m�t primitivn� funkce
k funkci~$f$, kter� je lich�, sud�, omezen�, periodick�, neklesaj�c�,
nez�porn�.

\nadpis{Racion�ln� funkce}

Racion�ln� funkce p�evedeme na parci�ln� zlomky. S t�mi pak nalo��me
podle n�sleduj�c�ho n�vodu.

$$
\begin{array}{ll}
\text{integrand}& \text{primitivn� funkce}\\
\frac{1}{x-\alpha}& \ln|x -\alpha|\\
\frac{1}{(x-\alpha)^k}; k > 1 & 
\frac{1}{-k+1}\frac{1}{(x-\alpha)^{k-1}}\\
\frac{2x + p}{x^2 + px + q}& \ln|x^2 + px + q|\\
\frac{1}{x^2 + px + q}& \frac{1}{\sqrt{q - 
\frac{p^2}{4}}}\text{arctg}\frac{x + \frac{p}{2}}{\sqrt{q - 
\frac{p^2}{4}}}\\
\frac{2x + p}{(x^2 + px + q)^k}; k > 1& \frac{1}{k+1}\frac{1}{(x^2 + 
px + q)^{k+1}}\\
\frac{1}{(x^2 + 1)^{k+1}}; k > 1&  \frac{1}{2k}\bigl( \frac{x}{(1 + x^2)^k}
+ 
(2k-1) \int \frac{1}{(1 + x^2)^k} \bigr)
\end{array}
$$


\pr 
\cast $\int \frac{3x+5}{2x^2+3x+7} \d x$, 
\cast $\int \frac{x^{17}-5}{x^2-1} \d x$, 
\cast $\int \frac{x^3+1}{x^3-5x^2+6x} \d x$, 
\cast $\int \frac{x^4}{x^4+5x^2+4} \d x$, 
\cast $\int \frac{x}{x^3-3x+2} \d x$, 
\cast $\int \frac{x}{x^3-1} \d x$, 
\cast $\int \frac{1}{x^6+1} \d x$, 
\cast $\int \frac{x}{2x^2-3x-2} \d x$, 
\cast $\int \frac{x^3+1}{x^3-x^2} \d x$, 
\cast $\int \frac{x^2}{(x+2)^2(x+4)^2} \d x$.

\end{document}

