Součásti dokumentu Matematika1Priklady
Zdrojový kód
%\wikiskriptum{Matematika1Priklady}
\section{Řešené Integrály}
%\begin{multicols}{2}
\begin{enumerate}
\begin{priklad}
\int \frac{\sqrt x -2\sqrt[3]{x^2} + 1}{\sqrt[4]{x}} dx= 4/5
x^{5/4} - 24/17 x ^{17/12} + 4/3 x ^{4/3}
\end{priklad}
\begin{priklad}
\int \frac{(1-x)^3}{x\sqrt[3]{x}}dx = 3x^{-1/3}(-1-3/2x+3/5x^2+1/8x^3)
\end{priklad}
\begin{priklad}
\int \frac{e^{3x} + 1}{e^x + 1}dx = \frac{e^{2x}}{2}-e^x+x
\end{priklad}
\begin{priklad}
\int (x + |x|)^2 dx = \frac{2}{3}(x^3 + |x^3|)
\end{priklad}
\begin{priklad}
\int \arccos(\cos x) dx; x \in (-\pi, \pi) = 1/2 x^2
\end{priklad}
\begin{priklad}
\int \arcsin(\sin x)dx; x \in (-\pi, \pi) = x^2/2 + \pi^2/4;
x \in \langle -\pi/2, \pi/3 \rangle; -x^2/2 + \pi|x|; x \in (
-\pi,-\pi/2\rangle \cup \langle \pi/2, \pi)
\end{priklad}
\begin{priklad}
\int \frac{x}{\sqrt{1+x^2 + \sqrt{(1+x^2)^3}}}dx =
2(1+\sqrt{1+x^2})^{1/2}
\end{priklad}
\begin{priklad}
\int \frac{x}{(x^2-1)^{3/2}}dx = -\frac{1}{\sqrt{x^2-1}}
\end{priklad}
\begin{priklad}
\int \frac{x}{4+x^4}dx = \frac{1}{4} \arctan \frac{x^2}{2}
\end{priklad}
\begin{priklad}
\int \frac{\sin x}{\sqrt{\cos^3 x}}dx = \frac{2}{\sqrt{\cos x}}
\end{priklad}
\begin{priklad}
\int x e^{-x^2}dx = -1/2 e ^{-x^2}
\end{priklad}
\begin{priklad}
\int \frac{\ln^2 x}{x}dx = \frac{\ln^3 x}{3}
\end{priklad}
\begin{priklad}
\int \frac{\ln x}{x\sqrt{1+\ln x}} = \frac{2}{3}\sqrt{1+\ln
x}(\ln x -2)
\end{priklad}
\begin{priklad}
\int \frac{1}{\sin^2 x} \frac{dx}{1+\tan x} = \ln |1+\cot x| -
\cot x
\end{priklad}
\begin{priklad}
\int \frac{\sin x \cos^3 x}{1+\cos^2 x} dx =
-\frac{1}{2}\cos^2x + \frac{1}{2} \ln(1+\cos^2x)
\end{priklad}
\begin{priklad}
\int \sqrt{x} \ln^2 x dx = \frac{2}{27}x^{3/2}(9 \ln^2x - 12 \ln x +8)
\end{priklad}
\begin{priklad}
\int x \sinh x dx = x \cosh x - \sinh x
\end{priklad}
\begin{priklad}
\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}
\end{priklad}
\begin{priklad}
\int \arctan \sqrt x dx = x \arctan \sqrt x + \arctan \sqrt x -
\sqrt x
\end{priklad}
\begin{priklad}
\int \frac{\ln \sin x}{\sin^2 x} = -\cot x \ln \sin x - \cot x
-x
\end{priklad}
\begin{priklad}
\int \limits_0^1 \arccos x dx = 1
\end{priklad}
\begin{priklad}
\int \limits_0^{2\pi}x^2 \cos x dx = 4 \pi
\end{priklad}
\begin{priklad}
\int \limits_0^{\sqrt 3}x \arctan x dx = \frac{2}{3}\pi -
\frac{\sqrt 3}{2}
\end{priklad}
\begin{priklad}
\int \limits_0^{\sqrt 3/2} \frac{x^5}{\sqrt{1-x^2}} =
\frac{53}{480}
\end{priklad}
\begin{priklad}
\int \frac{t}{(4t^2+9)^2} dt = -\frac{1}{8(4t^2 + 9)}
\end{priklad}
\begin{priklad}
\int \frac{b^3 x^3}{\sqrt{1-a^4x^4}}dx =
-\frac{b^3}{2a^4}\sqrt{1-a^4x^4}
\end{priklad}
\begin{priklad}
\int \limits_{-1}^1 \frac{r}{(1+r^2)^4}dr = 0
\end{priklad}
\begin{priklad}
\int \limits_0^a y \sqrt{a^2-y^2}dy = \frac{1}{3}|a|^3
\end{priklad}
\begin{priklad}
\int \limits_{-a}^0 y^2(1-\frac{y^3}{a^3})^{-2}dy = \frac{a^3}{6}
\end{priklad}
\begin{priklad}
\int \limits_0^1 \frac{x+3}{\sqrt{x+1}}dx = \frac{16}{3}\sqrt
2 - \frac{14}{3}
\end{priklad}
\begin{priklad}
\int \limits_{-1}^0x^3(x^2+1)^6 dx \footnote{subst $x^2 +1 =
u$}= -\frac{769}{112}
\end{priklad}
\begin{priklad}
\int x^{-1/2} \sin (x^{1/2}) dx = -2 \cos(x^{1/2})
\end{priklad}
\begin{priklad}
\int \sin^2 3x dx = \frac{1}{2}x-\frac{1}{12} \sin 6x
\end{priklad}
\begin{priklad}
\int \limits_0^{\pi/2} \sin^3 x \cos x dx = 1/4
\end{priklad}
\begin{priklad}
\int \limits_0^{2\pi} \cos^2 x dx = \pi
\end{priklad}
\begin{priklad}
\int \limits_0^1 \frac{\ln (x+1)}{x+1} dx = \frac{1}{2}(\ln 2 )^2
\end{priklad}
\begin{priklad}
\int \frac{\sqrt x}{1+x \sqrt{x} } dx= \frac{2}{3}\ln|1+x\sqrt x|
\end{priklad}
\begin{priklad}
\int \frac{e^{1/x}}{x^2} dx = - e ^{1/x}
\end{priklad}
\begin{priklad}
\int \limits_0^{\ln 2} \frac{e^x}{e^x + 1} dx = \ln \frac{3}{2}
\end{priklad}
\begin{priklad}
\int \frac{\log_2{x^3}}{x} dx = \frac{3}{\ln 4}(\ln x)^2
\end{priklad}
\begin{priklad}
\int \limits_1^2 2 ^{-x} dx = \frac{1}{4 \ln 2}
\end{priklad}
\begin{priklad}
\int \limits_{10}^{100} \frac{dx}{x \log_{10} x} = \ln 2 - \ln 10
\end{priklad}
\begin{priklad}
\int \limits_0^1 x 10^{1+x^2} dx = \frac{45}{ \ln 10}
\end{priklad}
\begin{priklad}
\int x e^{-x} dx = -x e^{-x} - e^{-x}
\end{priklad}
\begin{priklad}
\int x^2 e^{-x} dx = -e^{-x}(x^2+2x+2)
\end{priklad}
\begin{priklad}
\int \frac{x^2}{\sqrt{1-x}} = -2x^2(1-x)^{1/2} -
\frac{8}{3}x(1-x)^{3/2}-\frac{16}{15}(1-x)^{5/2}
\end{priklad}
\begin{priklad}
\int x \ln \sqrt x dx = \frac{1}{4} x^2 \ln x - \frac{1}{8}x^2
\end{priklad}
\begin{priklad}
\int \frac{\ln(x+1)}{\sqrt{x+1}} dx = 2\sqrt{x+1} \ln (x+1) -4
\sqrt{x+1}
\end{priklad}
\begin{priklad}
\int \ln^2 x dx = x \ln^2 x - 2x \ln x + 2x
\end{priklad}
\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})
\end{priklad}
\begin{priklad}
\int x^3 \sin x^2 dx = -\frac{1}{2}x^2 \cos x^2 + \frac{1}{2}
\sin x^2
\end{priklad}
\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{multicols}
\pagebreak