Matematika1Priklady:Kapitola6: Porovnání verzí

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%\wikiskriptum{Matematika1Priklady}
 
%\wikiskriptum{Matematika1Priklady}
\section{Řešené Integrály}
+
\section{Určité Integrály}
%\begin{multicols}{2}
+
 
 +
\subsection*{\fbox{Zkouškové příklady}}
 +
 
 +
 
 
\begin{enumerate}
 
\begin{enumerate}
  \begin{priklad}
+
 
    \int \frac{\sqrt x -2\sqrt[3]{x^2} + 1}{\sqrt[4]{x}} dx= 4/5
+
\item \begin{priklad}
    x^{5/4} - 24/17 x ^{17/12} + 4/3 x ^{4/3}
+
\int _{-1}^{1} 2x(x^2+1)\ud x
  \end{priklad}
+
\end{priklad}
+
\res{$0$}
  \begin{priklad}
+
 
    \int \frac{(1-x)^3}{x\sqrt[3]{x}}dx = 3x^{-1/3}(-1-3/2x+3/5x^2+1/8x^3)
+
\item \begin{priklad}
  \end{priklad}
+
\int_0^{\sqrt 3} \frac{2x^2 + 1}{\sqrt{3-x^2}}\ud x
+
\end{priklad}
  \begin{priklad}
+
\res{$2\pi $}
    \int \frac{e^{3x} + 1}{e^x + 1}dx = \frac{e^{2x}}{2}-e^x+x
+
 
  \end{priklad}
+
\item \begin{priklad}
+
\int_{-1}^1 \sqrt{3-x^2}\ud x
  \begin{priklad}
+
\end{priklad}
    \int (x + |x|)^2 dx = \frac{2}{3}(x^3 + |x^3|)
+
\res{$\sqrt {2}+3\,\arcsin \left( 1/3\,\sqrt {3} \right) $}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_1^2 \frac{6x^2-2}{x^3-x+1}\ud x
    \int \arccos(\cos x) dx; x \in (-\pi, \pi) = 1/2 x^2
+
\end{priklad}
  \end{priklad}
+
\res{$2\ln7 $}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \arcsin(\sin x)dx; x \in (-\pi, \pi) = x^2/2 + \pi^2/4;
+
\int_0^1 \arccos x \ud x  
    x \in \langle -\pi/2, \pi/3 \rangle; -x^2/2 + \pi|x|; x \in (
+
\end{priklad}
    -\pi,-\pi/2\rangle \cup \langle \pi/2, \pi)
+
\res{1}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_0^{2\pi}x^2 \cos x \ud x
    \int \frac{x}{\sqrt{1+x^2 + \sqrt{(1+x^2)^3}}}dx =
+
\end{priklad}
    2(1+\sqrt{1+x^2})^{1/2}
+
\res{$  4 \pi$}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_0^{\sqrt 3}x \arctan x \ud x
    \int \frac{x}{(x^2-1)^{3/2}}dx = -\frac{1}{\sqrt{x^2-1}}
+
\end{priklad}
  \end{priklad}
+
\res{$\frac{2}{3}\pi -\frac{\sqrt 3}{2} $}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \frac{x}{4+x^4}dx = \frac{1}{4} \arctan \frac{x^2}{2}
+
\int_0^{\sqrt 3/2} \frac{x^5}{\sqrt{1-x^2}}
  \end{priklad}
+
\end{priklad}
+
\res{$\frac{53}{480} $}
  \begin{priklad}
+
 
    \int \frac{\sin x}{\sqrt{\cos^3 x}}dx = \frac{2}{\sqrt{\cos x}}
+
 
  \end{priklad}
+
\item \begin{priklad}
+
\int_{-1}^1 \frac{r}{(1+r^2)^4} \ud r
  \begin{priklad}
+
\end{priklad}
    \int x e^{-x^2}dx = -1/2 e ^{-x^2}
+
\res{0}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_0^a y \sqrt{a^2-y^2}\ud y
    \int \frac{\ln^2 x}{x}dx = \frac{\ln^3 x}{3}
+
\end{priklad}
  \end{priklad}
+
\res{$\frac{1}{3}|a|^3$}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \frac{\ln x}{x\sqrt{1+\ln x}} = \frac{2}{3}\sqrt{1+\ln
+
\int_{-a}^0 y^2(1-\frac{y^3}{a^3})^{-2} \ud y
    x}(\ln x -2)
+
\end{priklad}
  \end{priklad}
+
\res{$\frac{a^3}{6} $}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \frac{1}{\sin^2 x} \frac{dx}{1+\tan x} = \ln |1+\cot x| -
+
\int_0^1 \frac{x+3}{\sqrt{x+1}}\ud x
    \cot x
+
\end{priklad}
  \end{priklad}
+
\res{$\frac{16}{3}\sqrt2 - \frac{14}{3} $}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \frac{\sin x \cos^3 x}{1+\cos^2 x} dx =
+
\int_{-1}^0x^3(x^2+1)^6 \ud x
    -\frac{1}{2}\cos^2x + \frac{1}{2} \ln(1+\cos^2x)
+
\end{priklad}
  \end{priklad}
+
\res{$-\frac{769}{112} $}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \sqrt{x} \ln^2 x dx = \frac{2}{27}x^{3/2}(9 \ln^2x - 12 \ln x +8)
+
\int_0^{\pi/2} \sin^3 x \cos x \ud x
  \end{priklad}
+
\end{priklad}
+
\res{$\frac14 $}
  \begin{priklad}
+
 
    \int x \sinh x dx = x \cosh x - \sinh x
+
\item \begin{priklad}
  \end{priklad}
+
\int_0^{2\pi} \cos^2 x \ud x
+
\end{priklad}
  \begin{priklad}
+
\res{$ \pi$}
    \int x^2 \arccos x dx = \frac{1}{3} x ^3 \arccos x +
+
 
    \frac{1}{9}(1-x^2)^{3/2} - \frac{1}{9}(1-x^2)^{1/2}
+
\item \begin{priklad}
  \end{priklad}
+
\int_0^1 \frac{\ln (x+1)}{x+1} \ud x
+
\end{priklad}
  \begin{priklad}
+
\res{$\frac{1}{2}(\ln 2 )^2 $}
    \int \arctan \sqrt x dx = x \arctan \sqrt x + \arctan \sqrt x -
+
 
    \sqrt x
+
\item \begin{priklad}
  \end{priklad}
+
\int_0^{\ln 2} \frac{e^x}{e^x + 1} \ud x
+
\end{priklad}
  \begin{priklad}
+
\res{$\ln \frac{3}{2} $}
    \int \frac{\ln \sin x}{\sin^2 x} = -\cot x \ln \sin x - \cot x
+
 
    -x
+
\item \begin{priklad}
  \end{priklad}
+
\int_1^2 2 ^{-x} \ud x
+
\end{priklad}
  \begin{priklad}
+
\res{$\frac{1}{4 \ln 2} $}
    \int \limits_0^1 \arccos x dx = 1
+
 
  \end{priklad}
+
\item \begin{priklad}
+
\int_{10}^{100} \frac{\ud x}{x \log_{10} x}  
  \begin{priklad}
+
\end{priklad}
    \int \limits_0^{2\pi}x^2 \cos x dx = 4 \pi
+
\res{$\ln 2 - \ln 10 $}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_0^1 x 10^{1+x^2} \ud x
    \int \limits_0^{\sqrt 3}x \arctan x dx = \frac{2}{3}\pi -
+
\end{priklad}
    \frac{\sqrt 3}{2}
+
\res{$\frac{45}{ \ln 10} $}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_0^{\ln \pi/4} e^x \frac{1}{\cos e^x} \ud x 
    \int \limits_0^{\sqrt 3/2} \frac{x^5}{\sqrt{1-x^2}} =
+
\end{priklad}
    \frac{53}{480}
+
\res{$\ln \Big((1+\sqrt 2)(\frac{1}{\cos 1} + \frac\pi4) \Big) $}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_0^5 \frac{\ud x}{25+x^2}
    \int \frac{t}{(4t^2+9)^2} dt = -\frac{1}{8(4t^2 + 9)}
+
\end{priklad}
  \end{priklad}
+
\res{$\frac{\pi}{20} $}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \frac{b^3 x^3}{\sqrt{1-a^4x^4}}dx =
+
\int_0^{3/2} \frac{\ud x}{9+4x^2}
    -\frac{b^3}{2a^4}\sqrt{1-a^4x^4}
+
\end{priklad}
  \end{priklad}
+
\res{$\frac{\pi}{24} $}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \limits_{-1}^1 \frac{r}{(1+r^2)^4}dr = 0
+
\int_{-3}^{-2} \frac{\ud x}{\sqrt{4-(x+3)^2}}  
  \end{priklad}
+
\end{priklad}
+
\res{$\frac{\pi}{6} $}
  \begin{priklad}
+
 
    \int \limits_0^a y \sqrt{a^2-y^2}dy = \frac{1}{3}|a|^3
+
\item \begin{priklad}
  \end{priklad}
+
\int_0^{\ln 2} \frac{e^x}{1+e^{2x}}\ud x
+
\end{priklad}
  \begin{priklad}
+
\res{$\arctan 2 -\frac{\pi}{4} $}
    \int \limits_{-a}^0 y^2(1-\frac{y^3}{a^3})^{-2}dy = \frac{a^3}{6}
+
 
  \end{priklad}
+
\item \begin{priklad}
+
\int_0^{\pi} \cos^4 x \ud x
  \begin{priklad}
+
\end{priklad}
    \int \limits_0^1 \frac{x+3}{\sqrt{x+1}}dx = \frac{16}{3}\sqrt
+
\res{$\frac38 $}
    2 - \frac{14}{3}
+
 
  \end{priklad}
+
\item \begin{priklad}
+
\int_0^{2\pi} \sin^3 x \cos x \ud x
  \begin{priklad}
+
\end{priklad}
    \int \limits_{-1}^0x^3(x^2+1)^6 dx \footnote{subst $x^2 +1 =
+
\res{$ 0$}
    u$}= -\frac{769}{112}
+
 
  \end{priklad}
+
\item \begin{priklad}
+
\int_0^{\frac32\pi} \cos^2 x \ud x
  \begin{priklad}
+
\end{priklad}
    \int x^{-1/2} \sin (x^{1/2}) dx = -2 \cos(x^{1/2})
+
\res{$\frac34\pi$}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_3^8 \frac{\ln x}{x} \ud x
    \int \sin^2 3x dx = \frac{1}{2}x-\frac{1}{12} \sin 6x
+
\end{priklad}
  \end{priklad}
+
\res{$-1/2\, \left( \ln  \left( 3 \right)  \right) ^{2}+9/2\, \left( \ln
+
  \left( 2 \right)  \right) ^{2}$}
  \begin{priklad}
+
 
    \int \limits_0^{\pi/2} \sin^3 x \cos x dx = 1/4
+
\item \begin{priklad}
  \end{priklad}
+
\int_0^{\ln 2} e^x \ud x
+
\end{priklad}
  \begin{priklad}
+
\res{$1$}
    \int \limits_0^{2\pi} \cos^2 x dx = \pi
+
 
  \end{priklad}
+
\item \begin{priklad}
+
\int_0^1 e^x(e^x+1)^{\frac15} \ud x
  \begin{priklad}
+
\end{priklad}
    \int \limits_0^1 \frac{\ln (x+1)}{x+1} dx = \frac{1}{2}(\ln 2 )^2
+
\res{$-(5/3)*2^(1/5)+(5/6)*(1+exp(1))^(6/5)$}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_{\pi/4}^{3/4\pi} \cotg x \ud x
    \int \frac{\sqrt x}{1+x \sqrt{x} } dx= \frac{2}{3}\ln|1+x\sqrt x|
+
\end{priklad}
  \end{priklad}
+
\res{0}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \frac{e^{1/x}}{x^2} dx = - e ^{1/x}
+
\int_0^{\pi/8} \frac{1}{\cos(2x)}\ud x
  \end{priklad}
+
\end{priklad}
+
\res{$-1/4\,\ln  \left( 2 \right) +1/2\,\ln \left( 2+\sqrt {2} \right) $}
  \begin{priklad}
+
 
    \int \limits_0^{\ln 2} \frac{e^x}{e^x + 1} dx = \ln \frac{3}{2}
+
\item \begin{priklad}
  \end{priklad}
+
\int _{-2}^1 \frac{x}{\sqrt{x^2+1}} \ud x
+
\end{priklad}
  \begin{priklad}
+
\res{$-\sqrt {5}+\sqrt {2}$}
    \int \frac{\log_2{x^3}}{x} dx = \frac{3}{\ln 4}(\ln x)^2
+
 
  \end{priklad}
+
\item \begin{priklad}
+
\int _{1}^{\sqrt2}x(x^2-1)^7 \ud x
  \begin{priklad}
+
\end{priklad}
    \int \limits_1^2 2 ^{-x} dx = \frac{1}{4 \ln 2}
+
\res{$\frac{1}{16}$}
  \end{priklad}
+
 
+
\item \begin{priklad}
  \begin{priklad}
+
\int_{-1}^{1} y (y+1)^{\frac12}dy
    \int \limits_{10}^{100} \frac{dx}{x \log_{10} x} = \ln 2 - \ln 10
+
\end{priklad}
  \end{priklad}
+
\res{$\frac4{15}\sqrt{2}$}
+
 
  \begin{priklad}
+
\item \begin{priklad}
    \int \limits_0^1 x 10^{1+x^2} dx = \frac{45}{ \ln 10}
+
\int_{0}^1 3x^2(x^3+1) \ud x
  \end{priklad}
+
\end{priklad}
+
\res{$\frac32$}
  \begin{priklad}
+
 
    \int x e^{-x} dx = -x e^{-x} - e^{-x}
+
\item \begin{priklad}
  \end{priklad}
+
\int_0^{2\pi} \cos^2 x \ud x
+
\end{priklad}
  \begin{priklad}
+
\res{$\pi$}
    \int x^2 e^{-x} dx = -e^{-x}(x^2+2x+2)
+
 
  \end{priklad}
+
 
+
\item
  \begin{priklad}
+
$\displaystyle \int_0^1 t^2(1-t^3)^8 \ud t$
    \int \frac{x^2}{\sqrt{1-x}} = -2x^2(1-x)^{1/2} -
+
\res{$\frac1{27} $}
    \frac{8}{3}x(1-x)^{3/2}-\frac{16}{15}(1-x)^{5/2}
+
 
  \end{priklad}
+
\item
+
$\displaystyle \int_0^1 \frac{r}{(1+r^2)^4} \ud r$
  \begin{priklad}
+
\res{$\frac{7}{48}$}
    \int x \ln \sqrt x dx = \frac{1}{4} x^2 \ln x - \frac{1}{8}x^2
+
 
  \end{priklad}
+
\item
+
$\displaystyle \int_0^{2\pi} \sin |x-\pi|~\ud x$.
  \begin{priklad}
+
\res{4}
    \int \frac{\ln(x+1)}{\sqrt{x+1}} dx = 2\sqrt{x+1} \ln (x+1) -4
+
 
    \sqrt{x+1}
+
\item
  \end{priklad}
+
$\displaystyle \int_0^1 x~ \arctg x~\ud x$.
+
\res{$\pi/4-1/2$}
  \begin{priklad}
+
 
    \int \ln^2 x dx = x \ln^2 x - 2x \ln x + 2x
+
\item
  \end{priklad}
+
$\displaystyle \int_0^{\pi} \sin( x+\pi)~\ud x$.
+
\res{-2}
  \begin{priklad}
+
 
    \int x^3 3 ^x dx = 3^x (\frac{x^3}{\ln 3} - \frac{3x^2}{\ln^2 3} + \frac{6x}{\ln^3 3} - \frac{6}{\ln^4 3})
+
\item
  \end{priklad}
+
$\displaystyle \int_0^1 e^x(e^x+1)^{1/5}\ud x$
+
\res{$5/6[(e+1)^{6/5} - 2^{6/5}]$}
  \begin{priklad}
+
 
    \int x^3 \sin x^2 dx = -\frac{1}{2}x^2 \cos x^2 + \frac{1}{2}
+
\item
    \sin x^2
+
$\displaystyle \int_0^1 \ln (x+1) \ud x$
  \end{priklad}
+
\res{$2\ln2-1$}
+
 
  \begin{priklad}
+
 
    \int \ln (1+x^2) dx = x \ln (1+x^2) - 2x +2 \arctan x
+
 
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \cot(\pi -x) dx
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \cot x \ln \sin x dx = \frac{1}{2}(\ln \sin x)^2
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \limits_0^{\ln \pi/4} e^x \frac{1}{\cos e^x} dx = \ln \Big((1+\sqrt 2)(\frac{1}{\cos 1} + \tan 1) \Big)
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \limits_0^5 \frac{dx}{25+x^2} = \frac{\pi}{20}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \limits_0^{3/2} \frac{dx}{9+4x^2} = \frac{\pi}{24}
+
  \end{priklad}
+
   
+
  \begin{priklad}
+
    \int \limits_{-3}^{-2} \frac{dx}{\sqrt{4-(x+3)^2}} = \frac{\pi}{6}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \limits_0^{\ln 2} \frac{e^x}{1+e^{2x}}dx = \arctan 2 -
+
    \frac{\pi}{4}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{\frac{1}{\cos^2 x}}{9+\tan^2 x}dx = \arctan(\frac{1}{3} \tan x)
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{\frac{1}{\cos^2 x}}{\sqrt{9-\tan^2 x}}dx
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \sinh^2( ax) \cosh( ax) dx = \frac{1}{3a}\sinh^3(ax)
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{\sinh ax}{\cosh ax}dx = \frac{1}{a} \ln(\cosh ax)
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{x^2}{\sqrt{4-x^2}}dx = 2 \arcsin(\frac{x}{2}) -
+
    \frac{1}{2}x\sqrt{4-x^2}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{x}{(1-x^2)^{3/2}}dx = \frac{1}{\sqrt{1-x^2}}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int x\sqrt{4-x^2}dx = -\frac{1}{3}(4-x^2)^{3/2}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{dx}{x\sqrt{a^2-x^2}} = \frac{1}{a} \ln \Big|\frac{a-\sqrt{a^2-x^2}}{x}\Big|
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{dx}{x^2\sqrt{a^2+x^2}} = -\frac{1}{a^2x}\sqrt{a^2+x^2}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{dx}{e^x\sqrt{e^{2x}-9}} = \frac{1}{9}e^{-x}\sqrt{e^{2x}-9}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int x \sqrt{6x-x^2-8} dx =
+
    -\frac{1}{3}(6x-x^2-8)^{3/2}+\frac{3}{2}\arcsin(x-3) +
+
    \frac{3}{2}\sqrt{6x-x^2-8}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{x}{(x^2+2x+5)^2}dx = \frac{x^2+x}{8(x^2+2x+5)} -
+
    \frac{1}{16}\arctan\big( \frac{x+1}{2} \big)
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{x+3}{\sqrt{x^2+4x+13}} = \sqrt{x^2+4x+13} + \ln(x+2+\sqrt{x^2+4x+13})
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \sqrt{6x-x^2-8}dx = \frac{1}{2}(x-3)\sqrt{6x-x^2-8} +
+
    \frac{1}{2} \arcsin(x-3)
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int x^2 \arcsin x dx = \frac{1}{3}x^3 \arcsin x +
+
    \frac{1}{3}(1-x^2)^{1/2} - \frac{1}{9}(1-x^2)^{3/2}
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{3}{\sqrt{2-3x-4x^2}}dx = \frac{3}{2} \arcsin \big(
+
    \frac{8x+3}{\sqrt{41}}
+
    \big)
+
  \end{priklad}
+
+
  \begin{priklad}
+
    \int \frac{x^2}{\sqrt{3-2x-x^2}}dx
+
  \end{priklad}
+
+
 
\end{enumerate}
 
\end{enumerate}
%\end{multicols}
 
 
\pagebreak
 

Verze z 10. 7. 2011, 11:48

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Součásti dokumentu Matematika1Priklady

součástakcepopisposlední editacesoubor
Hlavní dokument editovatHlavní stránka dokumentu Matematika1PrikladyFucikrad 18. 9. 201107:54
Řídící stránka editovatDefiniční stránka dokumentu a vložených obrázkůAdmin 7. 9. 201513:44
Header editovatHlavičkový souborPitrazby 23. 2. 201610:53 header.tex
Kapitola1 editovatLimity a spojitostPitrazby 25. 10. 201608:25 kapitola1.tex
Kapitola2 editovatDerivace, inverzní funkce, tečny, normály, asymptotyFucikrad 16. 7. 202011:14 kapitola2.tex
Kapitola3 editovatVyšetřování funkcíFucikrad 30. 11. 202011:43 kapitola3.tex
Kapitola4 editovatExtremální úlohy, konvexnost, konkávnost, inflexeFucikrad 12. 2. 201912:31 kapitola4.tex
Kapitola5 editovatNeurčité integrály a primitivní funkceFucikrad 18. 12. 201716:48 kapitola5.tex
Kapitola6 editovatUrčité integrályPitrazby 28. 4. 201611:29 kapitola6.tex
Kapitola7 editovatAplikace integrálůFucikrad 14. 12. 202017:33 kapitola7.tex

Zdrojový kód

%\wikiskriptum{Matematika1Priklady}
\section{Určité Integrály}
 
\subsection*{\fbox{Zkouškové příklady}}
 
 
\begin{enumerate}
 
\item \begin{priklad}
\int _{-1}^{1} 2x(x^2+1)\ud x
\end{priklad}
\res{$0$}
 
\item \begin{priklad}
\int_0^{\sqrt 3} \frac{2x^2 + 1}{\sqrt{3-x^2}}\ud x
\end{priklad}
\res{$2\pi $}
 
\item \begin{priklad}
\int_{-1}^1 \sqrt{3-x^2}\ud x
\end{priklad}
\res{$\sqrt {2}+3\,\arcsin \left( 1/3\,\sqrt {3} \right)  $}
 
\item \begin{priklad}
\int_1^2 \frac{6x^2-2}{x^3-x+1}\ud x
\end{priklad}
\res{$2\ln7 $}
 
\item \begin{priklad}
\int_0^1 \arccos x \ud x 
\end{priklad}
\res{1}
 
\item \begin{priklad}
\int_0^{2\pi}x^2 \cos x \ud x
\end{priklad}
\res{$  4 \pi$}
 
\item \begin{priklad}
\int_0^{\sqrt 3}x \arctan x \ud x 
\end{priklad}
\res{$\frac{2}{3}\pi -\frac{\sqrt 3}{2} $}
 
\item \begin{priklad}
\int_0^{\sqrt 3/2} \frac{x^5}{\sqrt{1-x^2}}
\end{priklad}
\res{$\frac{53}{480} $}
 
 
\item \begin{priklad}
\int_{-1}^1 \frac{r}{(1+r^2)^4} \ud r
\end{priklad}
\res{0}
 
\item \begin{priklad}
\int_0^a y \sqrt{a^2-y^2}\ud y
\end{priklad}
\res{$\frac{1}{3}|a|^3$}
 
\item \begin{priklad}
\int_{-a}^0 y^2(1-\frac{y^3}{a^3})^{-2} \ud y 
\end{priklad}
\res{$\frac{a^3}{6} $}
 
\item \begin{priklad}
\int_0^1 \frac{x+3}{\sqrt{x+1}}\ud x 
\end{priklad}
\res{$\frac{16}{3}\sqrt2 - \frac{14}{3} $}
 
\item \begin{priklad}
\int_{-1}^0x^3(x^2+1)^6 \ud x
\end{priklad}
\res{$-\frac{769}{112} $}
 
\item \begin{priklad}
\int_0^{\pi/2} \sin^3 x \cos x \ud x 
\end{priklad}
\res{$\frac14 $}
 
\item \begin{priklad}
\int_0^{2\pi} \cos^2 x \ud x
\end{priklad}
\res{$ \pi$}
 
\item \begin{priklad}
\int_0^1 \frac{\ln (x+1)}{x+1} \ud x 
\end{priklad}
\res{$\frac{1}{2}(\ln 2 )^2 $}
 
\item \begin{priklad}
\int_0^{\ln 2} \frac{e^x}{e^x + 1} \ud x 
\end{priklad}
\res{$\ln \frac{3}{2} $}
 
\item \begin{priklad}
\int_1^2 2 ^{-x} \ud x 
\end{priklad}
\res{$\frac{1}{4 \ln 2} $}
 
\item \begin{priklad}
\int_{10}^{100} \frac{\ud x}{x \log_{10} x} 
\end{priklad}
\res{$\ln 2 - \ln 10 $}
 
\item \begin{priklad}
\int_0^1 x 10^{1+x^2} \ud x 
\end{priklad}
\res{$\frac{45}{ \ln 10} $}
 
\item \begin{priklad}
\int_0^{\ln \pi/4} e^x \frac{1}{\cos e^x} \ud x  
\end{priklad}
\res{$\ln \Big((1+\sqrt 2)(\frac{1}{\cos 1} + \frac\pi4) \Big) $}
 
\item \begin{priklad}
\int_0^5 \frac{\ud x}{25+x^2} 
\end{priklad}
\res{$\frac{\pi}{20} $}
 
\item \begin{priklad}
\int_0^{3/2} \frac{\ud x}{9+4x^2}
\end{priklad}
\res{$\frac{\pi}{24} $}
 
\item \begin{priklad}
\int_{-3}^{-2} \frac{\ud x}{\sqrt{4-(x+3)^2}} 
\end{priklad}
\res{$\frac{\pi}{6} $}
 
\item \begin{priklad}
\int_0^{\ln 2} \frac{e^x}{1+e^{2x}}\ud x 
\end{priklad}
\res{$\arctan 2 -\frac{\pi}{4} $}
 
\item \begin{priklad}
\int_0^{\pi} \cos^4 x \ud x
\end{priklad}
\res{$\frac38 $}
 
\item \begin{priklad}
\int_0^{2\pi} \sin^3 x \cos x \ud x
\end{priklad}
\res{$ 0$}
 
\item \begin{priklad}
\int_0^{\frac32\pi} \cos^2 x \ud x
\end{priklad}
\res{$\frac34\pi$}
 
\item \begin{priklad}
\int_3^8 \frac{\ln x}{x} \ud x
\end{priklad}
\res{$-1/2\, \left( \ln  \left( 3 \right)  \right) ^{2}+9/2\, \left( \ln 
 \left( 2 \right)  \right) ^{2}$}
 
\item \begin{priklad}
\int_0^{\ln 2} e^x \ud x
\end{priklad}
\res{$1$}
 
\item \begin{priklad}
\int_0^1 e^x(e^x+1)^{\frac15} \ud x
\end{priklad}
\res{$-(5/3)*2^(1/5)+(5/6)*(1+exp(1))^(6/5)$}
 
\item \begin{priklad}
\int_{\pi/4}^{3/4\pi} \cotg x \ud x
\end{priklad}
\res{0}
 
\item \begin{priklad}
\int_0^{\pi/8} \frac{1}{\cos(2x)}\ud x
\end{priklad}
\res{$-1/4\,\ln  \left( 2 \right) +1/2\,\ln  \left( 2+\sqrt {2} \right) $}
 
\item \begin{priklad}
\int _{-2}^1 \frac{x}{\sqrt{x^2+1}} \ud x
\end{priklad}
\res{$-\sqrt {5}+\sqrt {2}$}
 
\item \begin{priklad}
\int _{1}^{\sqrt2}x(x^2-1)^7 \ud x
\end{priklad}
\res{$\frac{1}{16}$}
 
\item \begin{priklad}
\int_{-1}^{1} y (y+1)^{\frac12}dy
\end{priklad}
\res{$\frac4{15}\sqrt{2}$}
 
\item \begin{priklad}
\int_{0}^1 3x^2(x^3+1) \ud x
\end{priklad}
\res{$\frac32$}
 
\item \begin{priklad}
\int_0^{2\pi} \cos^2 x \ud x
\end{priklad}
\res{$\pi$}
 
 
\item 
$\displaystyle \int_0^1 t^2(1-t^3)^8 \ud t$
\res{$\frac1{27} $}
 
\item
$\displaystyle \int_0^1 \frac{r}{(1+r^2)^4} \ud r$
\res{$\frac{7}{48}$}
 
\item 
$\displaystyle \int_0^{2\pi} \sin |x-\pi|~\ud x$.
\res{4}
 
\item 
$\displaystyle \int_0^1 x~ \arctg x~\ud x$.
\res{$\pi/4-1/2$}
 
\item
$\displaystyle \int_0^{\pi} \sin( x+\pi)~\ud x$.
\res{-2}
 
\item 
$\displaystyle \int_0^1 e^x(e^x+1)^{1/5}\ud x$
\res{$5/6[(e+1)^{6/5} - 2^{6/5}]$}
 
\item
$\displaystyle \int_0^1 \ln (x+1) \ud x$
\res{$2\ln2-1$}
 
 
 
\end{enumerate}