\documentclass{article}
\usepackage{cvika}

\begin{document}

\Nadpis{\bf 4. cvi�en� z MA --- 18. a 19.3.2009}

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


\nadpis{P�evod na racion�ln� funkce}

N�kter� integr�ly lze p�ev�st na integraci racion�ln�ch funkc�.
V n�sleduj�c� tabulce zna�� $r(x,y)$  racion�ln� funkci dvou
prom�nn�ch s v�jimkou posledn�ch dvou ��dk�, kde $r$ z�vis� na jedn�
prom�nn�.

$$
\begin{array}{ll}
%\format
%\l\quad & \l\\
\text{typ integr�lu}&\text{doporu�en� substituce}  \\
\int r(x,\root n \of{\frac{ax + b}{cx + d}})\d x 
  & y = \phi(x) =\root n \of{\frac{ax + b}{cx + d}}\\
\int r(x, \sqrt{ax^2 + bx + c})\d x, a> 0 
  & \sqrt{ax^2 + bx + c} = \sqrt{a}x + y\\
\int r(x, \sqrt{ax^2 + bx + c})\d x, c> 0 
  & \sqrt{ax^2 + bx + c} = \sqrt{c} + yx\\
\int r(\sin x, \cos x) \d x 
  & y = \phi (x) = \text{tg}\frac{x}{2}\\
\int r(\sin x, \cos x) \d x, \quad \text{$r$ sud� v sinech a kosinech} 
  & y = \phi(x) = \text{tg}x\\ 
\int r(\sin x, \cos x) \d x, \quad\text{$r$ lich� v sinech} 
  & y = \phi(x) = \cos x\\
\int r(\sin x, \cos x) \d x, \quad\text{$r$ lich� v kosinech} 
  & y = \phi(x) = \sin x\\
\int r(e^x)\d x 
  & y = \phi(x) = e^x\\ 
\int r(\ln x). \frac{1}{x} \d x
  &  y = \phi(x) = \ln x
\end{array}
$$


\pr (substituce)

\cast $\int \sqrt\frac{1-x}{1+x} \d x$, 
\cast $\int \frac {1}{x} \root 3 \of{\frac{1-x}{1+x}} \d x$, 
\cast $\int \frac{x}{\sqrt{x+1} + \root 3 \of {x+1}} \d x$, 
\cast $\int \frac{1}{1+\sqrt{x+1}} \d x$, 
\cast $\int \frac{1}{\sqrt{x^2+1}} \d x$, 
\cast $\int \frac{1}{\sqrt{x^2-1}} \d x$, 
\cast $\int \frac{1}{\sqrt{(x^2-1)^3}} \d x$, 
\cast $\int \frac{1}{1+\sqrt{x^2+2x+2}} \d x$, 
\cast $\int \sqrt{x^2-2x-1} \d x$, 
\cast $\int \frac{2x+3}{(x^2+2x+3)\sqrt{x^2+2x+4}} \d x$. 

\pr (dal�� substituce)

\cast $\int \frac{1}{\sin x \cos x} \d x$, 
\cast $\int \frac{1}{\sin x} \d x$, 
\cast $\int \frac{\sin x}{1+\sin x} \d x$, 
\cast $\int \frac{1}{\cos x \sin^3 x} \d x$, 
\cast $\int \tg^5 x \d x$, 
\cast $\int \frac{\cos^4 x + \sin^4 x}{\cos^2 x - \sin^2 x} \d x$, 
\cast $\int \frac{1}{2\sin x - \cos x + 5} \d x$, 
\cast $\int \frac{\sin x \cos x}{1 + \sin^4 x} \d x$, 
\cast $\int \frac{e^x}{e^x + 1} \d x$, 
\cast $\int \frac{1}{x\sqrt {-\ln^2 x + 4\ln x - 3}} \d x$.

\pr (slo�it�j��)

\cast $\int \frac{x^2+1}{x^4+1} \d x$ ($t = x + 1/x$),
\cast $\int \sin \log x \d x$, 
\cast $\int \frac{1}{\sqrt {x-a}\sqrt{x-b}} \d x$
  ($x = a \cos^2 t + b \sin^2 t$),
\cast $\int \frac{1}{(x^2+1)^2} \d x$ ($x = \tg x$),
\cast $\int \sqrt{x^2+1} \d x$ ($x = \cosh t$),
\cast $\int \frac{\log\cos x}{\cos^2 x} \d x$, 
\cast $\int \sqrt {e^x - 1} \d x$, 
\cast $\int \frac{1}{\cos^3 x \sqrt{\sin 2x}} \d x$, 
\cast $\int \frac{1}{\sqrt{e^x+1}} \d x$, 
\cast $\int e^{\sqrt{x}} \d x$. 


\pr (lepen�)
\cast $\int |x| \d x$, 
\cast $\int \max\{x, x^2\} \d x$, 
\cast $\int |\sin x + \frac 12| \d x$.

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