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Mariusz M

Rejestracja: 11 Sep 2010
Offline Ostatnio: dziś, 05:41
*****

#129117 Rozwiąż równanie różnicowe

Napisane przez Mariusz M w 19.03.2017 - 04:58

y_{n+2}-4y_{n+1}+4y_{n}=0 \qquad y_{0}=3\\y_{1}=2\\</p>\\<p>y_{n}-4y_{n-1}+4y_{n-2}=0 \qquad y_{0}=3\\y_{1}=2\\</p>\\<p>y_{n}=4y_{n-1}-4y_{n-2}=0 \qquad y_{0}=3\\y_{1}=2\\</p>\\<p>Y\left(t\right)=\sum_{n=0}^{\infty}y_{n}t^n\\</p>\\<p>\sum_{n=2}^{\infty}y_{n}t^{n}=\sum_{n=2}^{\infty}4y_{n-1}t^n+\sum_{n=2}^{\infty}{-4y_{n-2}t^n}\\</p>\\<p>\sum_{n=2}^{\infty}y_{n}t^{n}=4t\sum_{n=2}^{\infty}y_{n-1}t^{n-1}-4t^2\sum_{n=2}^{\infty}{y_{n-2}t^{n-2}}\\</p>\\<p>\sum_{n=2}^{\infty}y_{n}t^{n}=4t\sum_{n=1}^{\infty}y_{n}t^{n}-4t^2\sum_{n=0}^{\infty}{y_{n}t^{n}}\\</p>\\<p>\sum_{n=0}^{\infty}y_{n}t^{n}-3-2t=4t\left(\sum_{n=0}^{\infty}y_{n}t^{n}-3\right)-4t^2\sum_{n=0}^{\infty}{y_{n}t^{n}}\\</p>\\<p>Y\left(t\right)-3-2t=4tY\left(t\right)-12t-4t^2Y\left(t\right)\\</p>\\<p>Y\left(t\right)\left(1-4t+4t^2\right)=-10t+3\\</p>\\<p>Y\left(t\right)=\frac{-10t+3}{\left(1-2t\right)^2}\\</p>\\<p>Y\left(t\right)=\frac{5-10t-2}{\left(1-2t\right)^2}\\</p>\\<p>Y\left(t\right)=\frac{5}{1-2t}-\frac{2}{\left(1-2t\right)^2}\\</p>\\<p>\sum_{n=0}^{\infty}2^{n}t^n=\frac{1}{1-2t}\\</p>\\<p>\frac{\mbox{d}}{\mbox{d}t}\left(\sum_{n=0}^{\infty}2^{n}t^n\right)=\frac{\mbox{d}}{\mbox{d}t}\left(\frac{1}{1-2t}\right)\\</p>\\<p>\sum_{n=0}^{\infty}n2^{n}t^{n-1}=-\frac{1}{\left(1-2t\right)^2}\cdot\left(-2\right)\\</p>\\<p>\sum_{n=1}^{\infty}n2^{n}t^{n-1}=\frac{2}{\left(1-2t\right)^2}\\</p>\\<p>\sum_{n=0}^{\infty}\left(n+1\right)2^{n+1}t^{n}=\frac{2}{\left(1-2t\right)^2}\\</p>\\<p>\sum_{n=0}^{\infty}\left(n+1\right)2^{n}t^{n}=\frac{1}{\left(1-2t\right)^2}\\</p>\\<p>Y\left(t\right)=\sum_{n=0}^{\infty}5\cdot 2^{n}t^{n}-2\sum_{n=0}^{\infty}\left(n+1\right)2^{n}t^n\\</p>\\<p>y_{n}=5\cdot 2^{n}-2\left(n+1\right)\cdot 2^{n}\\</p>\\<p>y_{n}=\left(3-2n\right)\cdot 2^{n}</p>\\<p>


  • 1


#129062 Całka 29

Napisane przez Mariusz M w 05.03.2017 - 06:11

Proponuję rozbić na dwie całki  w jednej licznik skróci się z mianownikiem a drugą można będzie policzyć całkując przez części

 

\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=-\frac{1}{198}\int{\frac{x^2+x+25}{\left(x^2+x+25\right)^2}\mbox{d}x}+\frac{1}{198}\int{\left(99x^2-49x+25\right)\cdot\frac{2x+1}{\left(x^2+x+25\right)^2}\mbox{d}x}\\</p>\\<p>\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=-\frac{1}{198}\int{\frac{1}{\left(x^2+x+25\right)}\mbox{d}x}+\frac{1}{198}\int{\left(99x^2-49x+25\right)\cdot\frac{2x+1}{\left(x^2+x+25\right)^2}\mbox{d}x}\\</p>\\<p>

 

Najpierw całkujemy przez części a później dodajemy całki

 

Przydatne będzie też podstawienie x+\frac{1}{2}=\frac{3}{2}\sqrt{11}t

 

Jak znalazłem ten rozkład ?

 

Podzieliłem licznik przez pochodną trójmianu kwadratowego z mianownika

Otrzymałem  resztę która była stałą, do tej stałej dodałem i odjąłem taki składnik aby otrzymać postać kanoniczną trójmianu kwadratowego z mianownika

 

\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=-\frac{1}{198}\int{\frac{x^2+x+25}{\left(x^2+x+25\right)^2}\mbox{d}x}+\frac{1}{198}\int{\left(99x^2-49x+25\right)\cdot\frac{2x+1}{\left(x^2+x+25\right)^2}\mbox{d}x}\\</p>\\<p>\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=-\frac{1}{198}\int{\frac{1}{\left(x^2+x+25\right)}\mbox{d}x}+\frac{1}{198}\int{\left(99x^2-49x+25\right)\cdot\frac{2x+1}{\left(x^2+x+25\right)^2}\mbox{d}x}\\</p>\\<p>\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=-\frac{1}{198}\int{\frac{1}{\left(x^2+x+25\right)}\mbox{d}x}-\frac{1}{198}\frac{99x^2-49x+25}{x^2+x+25}+\frac{1}{198}\int{\frac{198x-49}{x^2+x+25}\mbox{d}x}\\</p>\\<p>\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=-\frac{1}{198}\frac{99x^2-49x+25}{x^2+x+25}+\frac{1}{99}\int{\frac{99x-25}{x^2+x+25}\mbox{d}x}\\</p>\\<p>\frac{1}{99}\int{\frac{99x-25}{x^2+x+25}\mbox{d}x}=\frac{1}{99}\int{\frac{99x-25}{\left(x+\frac{1}{2}\right)^2+\frac{99}{4}}\mbox{d}x}\\</p>\\<p>x+\frac{1}{2}=\frac{3}{2}\sqrt{11}t\\</p>\\<p>\mbox{d}x=\frac{3}{2}\sqrt{11}\mbox{d}t\\</p>\\<p>\frac{\sqrt{11}}{66}\int{\frac{99\left(\frac{3}{2}\sqrt{11}t-\frac{1}{2}\right)-25}{\frac{99}{4}t^2+\frac{99}{4}}\mbox{d}t}\\</p>\\<p>\frac{\sqrt{11}}{66}\int{\frac{\frac{297}{2}\sqrt{11}t-\frac{149}{2}}{\frac{99}{4}t^2+\frac{99}{4}}\mbox{d}t}\\</p>\\<p>\frac{\sqrt{11}}{4 \cdot 33}\cdot\frac{4}{99}\int{\frac{297\sqrt{11}t-149}{t^2+1}\mbox{d}t}\\</p>\\<p>\frac{297\cdot 11}{3267}\int{\frac{t}{t^2+1}\mbox{d}t}-\frac{149\sqrt{11}}{3267}\int{\frac{1}{t^2+1}\mbox{d}t}\\</p>\\<p>=\int{\frac{t}{t^2+1}\mbox{d}t}-\frac{149\sqrt{11}}{3267}\int{\frac{1}{t^2+1}\mbox{d}t}\\</p>\\<p>=\frac{1}{2}\ln{\left|t^2+1\right|}-\frac{149\sqrt{11}}{3267}\arctan{\left(t\right)}+C\\</p>\\<p>\frac{1}{2}\ln{\left|x^2+x+25\right|}-\frac{149\sqrt{11}}{3267}\arctan{\left(\frac{\sqrt{11}}{33}\left(2x+1\right)\right)}+C\\</p>\\<p>\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=-\frac{1}{198}\frac{99x^2-49x+25}{x^2+x+25}+\frac{1}{2}\ln{\left|x^2+x+25\right|}-\frac{149\sqrt{11}}{3267}\arctan{\left(\frac{\sqrt{11}}{33}\left(2x+1\right)\right)}+C\\</p>\\<p>

 

 

Wynik można jeszcze uprościć dzieląc licznik przez mianownik

 

 

Można też liczyć sposobem podanym przez Ostrogradskiego

 

\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=\frac{a_{1}x+a_{0}}{x^2+x+25}+\int{\frac{b_{1}x+b_{0}}{x^2+x+25}\mbox{d}x}\\</p>\\<p>\frac{x^3}{\left(x^2+x+25\right)^2}=\frac{a_{1}\left(x^2+x+25\right)-\left(a_{1}x+a_{0}\right)\left(2x+1\right)}{\left(x^2+x+25\right)^2}+\frac{b_{1}x+b_{0}}{x^2+x+25}\\</p>\\<p>\frac{x^3}{\left(x^2+x+25\right)^2}=\frac{a_{1}\left(x^2+x+25\right)-\left(a_{1}x+a_{0}\right)\left(2x+1\right)+\left(b_{1}x+b_{0}\right)\left(x^2+x+25\right)}{\left(x^2+x+25\right)^2}\\</p>\\<p>x^3=a_{1}\left(x^2+x+25\right)-\left(a_{1}x+a_{0}\right)\left(2x+1\right)+\left(b_{1}x+b_{0}\right)\left(x^2+x+25\right)\\</p>\\<p>x^3=a_{1}x^2+a_{1}x+25a_{1}-2a_{1}x^2-a_{1}x-2a_{0}x-a_{0}+b_{1}x^3+b_{1}x^2+25b_{1}x+b_{0}x^2+b_{0}x+25b_{0}\\</p>\\<p>x^3=b_{1}x^3+\left(b_{1}+b_{0}-a_{1}\right)x^2+\left(25b_{1}+b_{0}-2a_{0}\right)x+25b_{0}+25a_{1}-a_{0}\\</p>\\<p>\begin{cases}b_{1}=1\\b_{0}=a_{1}-1\\24+a_{1}=2a_{0}\\100a_{1}-50-24-a_{1}=0\end{cases}\\</p>\\<p>\begin{cases}b_{1}=1\\\b_{0}=a_{1}-1\\24+a_{1}=2a_{0}\\99a_{1}=74\end{cases}\\</p>\\<p>\begin{cases}b_{1}=1\\99b_{0}=-25\\99a_{0}=1225\\99a_{1}=74\end{cases}\\</p>\\<p>\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=\frac{1}{99}\frac{74x+1225}{x^2+x+25}+\frac{1}{99}\int{\frac{99x-25}{x^2+x+25}\mbox{d}x}\\</p>\\<p>\frac{1}{99}\int{\frac{99x-25}{x^2+x+25}\mbox{d}x}=\frac{1}{99}\int{\frac{99x-25}{\left(x+\frac{1}{2}\right)^2+\frac{99}{4}}\mbox{d}x}\\</p>\\<p>x+\frac{1}{2}=\frac{3}{2}\sqrt{11}t\\</p>\\<p>\mbox{d}x=\frac{3}{2}\sqrt{11}\mbox{d}t\\</p>\\<p>\frac{\sqrt{11}}{66}\int{\frac{99\left(\frac{3}{2}\sqrt{11}t-\frac{1}{2}\right)-25}{\frac{99}{4}t^2+\frac{99}{4}}\mbox{d}t}\\</p>\\<p>\frac{\sqrt{11}}{66}\int{\frac{\frac{297}{2}\sqrt{11}t-\frac{149}{2}}{\frac{99}{4}t^2+\frac{99}{4}}\mbox{d}t}\\</p>\\<p>\frac{\sqrt{11}}{4 \cdot 33}\cdot\frac{4}{99}\int{\frac{297\sqrt{11}t-149}{t^2+1}\mbox{d}t}\\</p>\\<p>\frac{297\cdot 11}{3267}\int{\frac{t}{t^2+1}\mbox{d}t}-\frac{149\sqrt{11}}{3267}\int{\frac{1}{t^2+1}\mbox{d}t}\\</p>\\<p>=\int{\frac{t}{t^2+1}\mbox{d}t}-\frac{149\sqrt{11}}{3267}\int{\frac{1}{t^2+1}\mbox{d}t}\\</p>\\<p>=\frac{1}{2}\ln{\left|t^2+1\right|}-\frac{149\sqrt{11}}{3267}\arctan{\left(t\right)}+C\\</p>\\<p>\frac{1}{2}\ln{\left|x^2+x+25\right|}-\frac{149\sqrt{11}}{3267}\arctan{\left(\frac{\sqrt{11}}{33}\left(2x+1\right)\right)}+C\\</p>\\<p>\int{\frac{x^3}{\left(x^2+x+25\right)^2}\mbox{d}x}=\frac{1}{99}\frac{74x+1225}{x^2+x+25}+\frac{1}{2}\ln{\left|x^2+x+25\right|}-\frac{149\sqrt{11}}{3267}\arctan{\left(\frac{\sqrt{11}}{33}\left(2x+1\right)\right)}+C\\</p>\\<p>


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#129051 Całka 24

Napisane przez Mariusz M w 03.03.2017 - 08:31

\int{\frac{\mbox{d}x}{\left(x^2+x+1\right)\sqrt{x^2+x-1}}}\\</p>\\<p>\sqrt{x^2+x-1}=t-x\\</p>\\<p>x^2+x-1=t^2-2tx+x^2\\</p>\\<p>x-1=t^2-2tx\\</p>\\<p>2tx+x=t^2+1\\</p>\\<p>x\left(2t+1\right)=t^2+1\\</p>\\<p>x=\frac{t^2+1}{2t+1}\\</p>\\<p>t-x=\frac{2t^2+t-t^2-1}{2t+1}=\frac{t^2+t-1}{2t+1}\\</p>\\<p>x^2+x+1=\frac{t^4+2t^2+1+2t^3+t^2+2t+1+4t^2+4t+1}{\left(2t+1\right)^2}\\<br>\\x^2+x+1=\frac{t^4+2t^3+7t^2+6t+3}{\left(2t+1\right)^2}\\</p>\\<p>\mbox{d}x=\frac{2t\left(2t+1\right)-2\left(t^2+1\right)}{\left(2t+1\right)^2}\mbox{d}t\\</p>\\<p>\mbox{d}x=\frac{2t^2+2t-2}{\left(2t+1\right)^2}\mbox{d}t\\</p>\\<p>\int{\frac{\left(2t+1\right)^2}{t^4+2t^3+7t^2+6t+3}\cdot\frac{2t+1}{t^2+t-1}\cdot\frac{2\left(t^2+t-1\right)}{\left(2t+1\right)^2}\mbox{d}t}\\</p>\\<p>\int{\frac{4t+2}{t^4+2t^3+7t^2+6t+3}\mbox{d}t}=\int{\frac{4t+2}{\left(t^2+t+3-\sqrt{6}\right)\left(t^2+t+3+\sqrt{6}\right)}\mbox{d}t}\\</p>\\<p>=\int{\frac{\sqrt{6}}{6}\left(\frac{2t+1}{t^2+t+3-\sqrt{6}}-\left(\frac{2t+1}{t^2+t+3-\sqrt{6}}\right)\right)\mbox{d}t}\\</p>\\<p>=\frac{\sqrt{6}}{6}\int{\frac{2t+1}{t^2+t+3-\sqrt{6}}\mbox{d}t}-\frac{\sqrt{6}}{6}\int{\frac{2t+1}{t^2+t+3+\sqrt{6}}\mbox{d}t}\\</p>\\<p>=\frac{\sqrt{6}}{6}\ln{\left|\frac{t^2+t+3-\sqrt{6}}{t^2+t+3+\sqrt{6}}\right|}+C</p>\\<p>


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#128996 całka 28

Napisane przez Mariusz M w 28.02.2017 - 05:35

\int{\frac{\mbox{d}x}{\left(x^2-1\right)^2}e^{x}\left(1+2x-x^2\right)}\\<br>\\\int{\frac{1-x^2}{\left(x^2-1\right)^2}e^{x}\mbox{d}x}+\int{\frac{2x}{\left(x^2-1\right)^2}e^{x}\mbox{d}x}\\<br>\\=-\int{\frac{e^{x}}{x^2-1}\mbox{d}x}+\int{\frac{2x}{\left(x^2-1\right)}e^{x}\mbox{d}x}\\<br>\\=-\int{\frac{e^{x}}{x^2-1}\mbox{d}x}-\frac{e^{x}}{x^2-1}+\int{\frac{e^{x}}{x^2-1}\mbox{d}x}\\<br>\\=-\frac{e^{x}}{x^2-1}+C


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#128938 Całka wymierna, Ostrogradskiego

Napisane przez Mariusz M w 21.02.2017 - 22:02

To podstawienie jest kiepskie pod względem metodyki nauczania bo całki z funkcji wymiernej poznajemy wcześniej

ale za to jakie modne amerykańskie

 

\int{\frac{3x+5}{\left(x^2+1\right)^3}\mbox{d}x}=\frac{a_{3}x^3+a_{2}x^2+a_{1}x+a_{0}}{\left(x^2+1\right)^2}+\int{\frac{b_{1}x+b_{0}}{x^2+1}\mbox{d}x}\\</p>\\<p>\frac{3x+5}{\left(x^2+1\right)^3}=\frac{\left(3a_{3}x^2+2a_{2}x+a_{1}\right)\left(x^2+1\right)^2-\left(a_{3}x^3+a_{2}x^2+a_{1}x+a_{0}\right)\left(1+x^2\right)\cdot 4x}{\left(1+x^2\right)^4}+\frac{b_{1}x+b_{0}}{x^2+1}\\</p>\\<p>\frac{3x+5}{\left(x^2+1\right)^3}=\frac{\left(3a_{3}x^2+2a_{2}x+a_{1}\right)\left(x^2+1\right)-\left(a_{3}x^3+a_{2}x^2+a_{1}x+a_{0}\right)\cdot 4x+\left(b_{1}x+b_{0}\right)\left(x^4+2x+1\right)}{\left(x^2+1\right)^3}\\</p>\\<p>3x+5=\left(3a_{3}x^2+2a_{2}x+a_{1}\right)\left(x^2+1\right)-\left(a_{3}x^3+a_{2}x^2+a_{1}x+a_{0}\right)\cdot 4x+\left(b_{1}x+b_{0}\right)\left(x^4+2x+1\right)\\</p>\\<p>3x+5=3a_{3}x^4+2a_{2}x^3+a_{1}x^2+3a_{3}x^2+2a_{2}x+a_{1}-4a_{3}x^{4}-4a_{2}x^{3}-4a_{1}x^2-4a_{0}x+b_{1}x^5+2b_{1}x^3+b_{1}x+b_{0}x^4+2b_{0}x^2+b_{0}\\</p>\\<p>3x+5=b_{1}x^5+\left(b_{0}-a_{3}\right)x^4+\left(2b_{1}-2a_{2}\right)x^3+\left(2b_{0}+3a_{3}-3a_{1}\right)x^2+\left(b_{1}+2a_{2}-4a_{0}\right)x+a_{1}+b_{0}\\</p>\\<p>\begin{cases}b_{1}=0\\b_{0}-a_{3}=0\\2b_{1}-2a_{2}=0\\2b_{0}+3a_{3}-3a_{1}=0\\b_{1}+2a_{2}-4a_{0}=3\\a_{1}+b_{0}=5\end{cases}\\</p>\\<p>\begin{cases}b_{1}=0\\b_{0}=a_{3}\\a_{2}=0\\5a_{3}-3a_{1}=0\\-4a_{0}=3\\a_{3}+a_{1}=5\end{cases}\\</p>\\<p>\begin{cases}b_{1}=0\\b_{0}=a_{3}\\a_{2}=0\\5a_{3}-3a_{1}=0\\-4a_{0}=3\\3a_{3}+3a_{1}=15\end{cases}\\</p>\\<p>\begin{cases}b_{1}=0\\b_{0}=a_{3}\\a_{2}=0\\3a_{1}=5a_{3}\\-4a_{0}=3\\8a_{3}=15\end{cases}\\</p>\\<p>\begin{cases}b_{1}=0\\b_{0}=\frac{15}{8}\\a_{3}=\frac{15}{8}\\a_{2}=0\\a_{1}=\frac{25}{8}\\a_{0}=-\frac{6}{8}\end{cases}\\</p>\\<p>\int{\frac{3x+5}{\left(x^2+1\right)^3}\mbox{d}x}=\frac{1}{8}\frac{15x^3+25x-6}{\left(x^2+1\right)^2}+\frac{15}{8}\arctan{\left(x\right)}+C</p>\\<p>

 

 

 


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#128919 Całka 19

Napisane przez Mariusz M w 19.02.2017 - 23:34

\sqrt{4-x^2}=\left(2+x\right)t\\</p>\\<p>\left(2+x\right)\left(2-x\right)=\left(2+x\right)^2t^2</p>\\<p>2-x=\left(2+x\right)t^2</p>\\<p>2-x=2t^2+xt^2</p>\\<p>x+xt^2=2-2t^2</p>\\<p>x\left(1+t^2\right)=2-2t^2</p>\\<p>x=\frac{2-2t^2}{1+t^2}=\frac{-2-2t^2+4}{1+t^2}=-2+\frac{4}{1+t^2}\\</p>\\<p>\left(2+x\right)t=\frac{4t}{1+t^2}\\</p>\\<p>\mbox{d}x=-4\left(1+t^2\right)^{-2}\cdot 2t\mbox{d}t\\</p>\\<p>\mbox{d}x=-\frac{8t}{\left(1+t^2\right)^{2}}\mbox{d}t\\</p>\\<p>\int{\frac{\left(1+t^2\right)^6}{64\left(1-t^2\right)^6}\cdot\frac{64t^3}{\left(1+t^2\right)^3}\cdot\left(-\frac{8t}{\left(1+t^2\right)^{2}}\right)\mbox{d}t}\\</p>\\<p>-8\int{\frac{t^4\left(1+t^2\right)}{\left(1-t^2\right)^6}\mbox{d}t}\\</p>\\<p>\int{\left(-4t^3-4t^5\right)\frac{2t}{\left(1-t^2\right)^6}\mbox{d}t}=-\frac{4}{5}\frac{t^3+t^5}{\left(1-t^2\right)^5}+\frac{4}{5}\int{\frac{3t^2+5t^4}{\left(1-t^2\right)^5}\mbox{d}t}</p>\\<p>-8\int{\frac{t^4\left(1+t^2\right)}{\left(1-t^2\right)^6}\mbox{d}t}=-\frac{4}{5}\frac{t^3+t^5}{\left(1-t^2\right)^5}+\frac{2}{5}\int{\left(3t+5t^3\right)\frac{2t}{\left(1-t^2\right)^5}\mbox{d}t}\\</p>\\<p>-8\int{\frac{t^4\left(1+t^2\right)}{\left(1-t^2\right)^6}\mbox{d}t}=-\frac{4}{5}\frac{t^3+t^5}{\left(1-t^2\right)^5}+\frac{1}{10}\frac{3t+5t^3}{\left(1-t^2\right)^4}-\frac{3}{10}\int{\frac{1+5t^2}{\left(1-t^2\right)^4}\mbox{d}t}\\</p>\\<p>-8\int{\frac{t^4\left(1+t^2\right)}{\left(1-t^2\right)^6}\mbox{d}t}=-\frac{4}{5}\frac{t^3+t^5}{\left(1-t^2\right)^5}+\frac{1}{10}\frac{3t+5t^3}{\left(1-t^2\right)^4}-\frac{3}{10}\int{\frac{1-t^2+6t^2}{\left(1-t^2\right)^4}\mbox{d}t}\\</p>\\<p>-8\int{\frac{t^4\left(1+t^2\right)}{\left(1-t^2\right)^6}\mbox{d}t}=-\frac{4}{5}\frac{t^3+t^5}{\left(1-t^2\right)^5}+\frac{1}{10}\frac{3t+5t^3}{\left(1-t^2\right)^4}-\frac{3}{10}\int{\frac{\mbox{d}t}{\left(1-t^2\right)^3}\mbox{d}t}-\frac{9}{10}\int{t\frac{2t}{\left(1-t^2\right)^4}\mbox{d}t}\\</p>\\<p></p>\\<p>-8\int{\frac{t^4\left(1+t^2\right)}{\left(1-t^2\right)^6}\mbox{d}t}=-\frac{4}{5}\frac{t^3+t^5}{\left(1-t^2\right)^5}+\frac{1}{10}\frac{3t+5t^3}{\left(1-t^2\right)^4}-\frac{3}{10}\int{\frac{\mbox{d}t}{\left(1-t^2\right)^3}\mbox{d}t}-\frac{3}{10}\frac{t}{\left(1-t^2\right)^3}+\frac{3}{10}\int{\frac{\mbox{d}t}{\left(1-t^2\right)^3}}\\</p>\\<p>-8\int{\frac{t^4\left(1+t^2\right)}{\left(1-t^2\right)^6}\mbox{d}t}=-\frac{4}{5}\frac{t^3+t^5}{\left(1-t^2\right)^5}+\frac{1}{10}\frac{3t+5t^3}{\left(1-t^2\right)^4}-\frac{3}{10}\frac{t}{\left(1-t^2\right)^3}\\<br>\\


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#128916 Całka 37

Napisane przez Mariusz M w 19.02.2017 - 18:30

Jeśli chcesz sprowadzić do całki z funkcji wymiernej to

 

</p>\\<p>-\left(x+2-\sqrt{5}\right)\left(x+2+\sqrt{5}\right)\\</p>\\<p>\sqrt{-\left(x+2-\sqrt{5}\right)\left(x+2+\sqrt{5}\right)}=\left(x+2+\sqrt{5}\right)t\\</p>\\<p>-\left(x+2-\sqrt{5}\right)\left(x+2+\sqrt{5}\right)=\left(x+2+\sqrt{5}\right)^2t^2\\</p>\\<p>-\left(x+2-\sqrt{5}\right)=\left(x+2+\sqrt{5}\right)t^2\\</p>\\<p>-x-2+\sqrt{5}=xt^2+\left(2+\sqrt{5}\right)t^2\\</p>\\<p>-x-xt^2=\left(2+\sqrt{5}\right)t^2+2-\sqrt{5}\\</p>\\<p>x\left(1+t^2\right)=-\left(2+\sqrt{5}\right)t^2-2+\sqrt{5}\\</p>\\<p>x=-\frac{\left(2+\sqrt{5}\right)t^2+2-\sqrt{5}}{1+t^2}=-\frac{\left(2+\sqrt{5}\right)t^2+2+\sqrt{5}-2\sqrt{5}}{1+t^2}=-2-\sqrt{5}+2\sqrt{5}\frac{1}{\left(1+t^2\right)}\\</p>\\<p>\left(x+2+\sqrt{5}\right)t=2\sqrt{5}\frac{t}{\left(1+t^2\right)}\\</p>\\<p>\mbox{d}x=-2\sqrt{5}\cdot\left(t^2+1\right)^{-2}\cdot 2t\mbox{d}t\\</p>\\<p>\mbox{d}x=-4\sqrt{5}\frac{t}{\left(t^2+1\right)^2}\mbox{d}t\\</p>\\<p>\int{\left(-\frac{\left(2+\sqrt{5}\right)t^2+2-\sqrt{5}}{1+t^2}-1\right)\cdot\frac{1+t^2}{2\sqrt{5}t}\cdot\left(-4\sqrt{5}\frac{t}{\left(t^2+1\right)^2}\right)\mbox{d}t}</p>\\<p>

 

Jeśli nie chcesz sprowadzać całki do całki z funkcji wymiernej to

możesz zapisać trójmian kwadratowy pod pierwiastkiem w postaci kanonicznej

 


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#128184 całka10

Napisane przez Mariusz M w 01.10.2016 - 00:53

\int{\frac{x^2}{\sqrt{13+11x^2}}\mbox{d}x}\\</p>\\<p>\sqrt{13+11x^2}=t-\sqrt{11}x\\</p>\\<p>13+11x^2=t^2-2\sqrt{11}tx+11x^2\\</p>\\<p>13=t^2-2\sqrt{11}tx\\</p>\\<p>2\sqrt{11}tx=t^2-13</p>\\<p>x=\frac{t^2-13}{2\sqrt{11}t}\\</p>\\<p>\mbox{d}x=\frac{2t\cdot 2\sqrt{11}t-2\sqrt{11}\left(t^2-13\right)}{44t^2}\mbox{d}t\\</p>\\<p>\mbox{d}x=\sqrt{11}\frac{t^2+13}{22t^2}\mbox{d}t\\</p>\\<p>t-\sqrt{11}x=\frac{2\sqrt{11}t^2-\sqrt{11}t^2+13\sqrt{11}}{2\sqrt{11}t}\\</p>\\<p>t-\sqrt{11}x=\frac{t^2+13}{2t}\\</p>\\<p>\int{\frac{\left(t^2-13\right)^2}{44t^2}\cdot\frac{2t}{t^2+13}\cdot\sqrt{11}\frac{t^2+13}{22t^2}\mbox{d}t}\\</p>\\<p>\frac{\sqrt{11}}{484}\int{\frac{\left(t^2-13\right)^2}{t^3}\mbox{d}t}\\</p>\\<p>\frac{\sqrt{11}}{484}\int{\frac{t^4-26t^2+169}{t^3}\mbox{d}t}\\</p>\\<p>\frac{\sqrt{11}}{484}\left(\int{t\mbox{d}t}+169\int{\frac{\mbox{d}t}{t^3}}-26\int{\frac{\mbox{d}t}{t}}\right)\\</p>\\<p>\frac{\sqrt{11}}{484}\left(\frac{t^2}{2}-\frac{169}{2}\frac{1}{t^2}-26\ln{\left|t\right|}\right)+C\\</p>\\<p>\frac{\sqrt{11}}{242}\left(\frac{t^4-169}{4t^2}-13\ln{\left|t\right|}\right)+C\\</p>\\<p>\frac{\sqrt{11}}{242}\left(\frac{\left(t^2-13\right)\left(t^2+13\right)}{4t^2}-13\ln{\left|t\right|}\right)+C\\</p>\\<p>\frac{\sqrt{11}}{242}\left(\sqrt{11}x\sqrt{11x^2+13}-13\ln{\left|\sqrt{11}x+\sqrt{11x^2+13}\right|}\right)+C\\</p>\\<p>

 


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#127944 całka12

Napisane przez Mariusz M w 13.08.2016 - 04:32

\int{\frac{\mbox{d}x}{\sqrt{x^2-2x}}}\\</p>\\<p>\sqrt{x^2-2x}=t-x\\</p>\\<p>x^2-2x=t^2-2tx+x^2</p>\\<p>-2x=t^2-2tx\\</p>\\<p>2tx-2x=t^2</p>\\<p>x\left(2t-2\right)=t^2\\</p>\\<p>x=\frac{t^2}{2t-2}\\</p>\\<p>t-x=\frac{2t^2-2t-t^2}{2t-2}=\frac{t^2-2t}{2t-2}\\</p>\\<p>\mbox{d}x=\frac{2t\left(2t-2\right)-2t^2}{\left(2t-2\right)^2}\mbox{d}t\\</p>\\<p>\mbox{d}x=\frac{t^2-2t}{2\left(t-1\right)^2}\mbox{d}t\\</p>\\<p>\int{\frac{2\left(t-1\right)}{t^2-2t}\cdot\frac{t^2-2t}{2\left(t-1\right)^2}\mbox{d}t}\\</p>\\<p>\int{\frac{\mbox{d}t}{t-1}}\\</p>\\<p>\ln{\left|t-1\right|}+C\\</p>\\<p>\ln{\left|x-1+\sqrt{x^2-2x}\right|}+C\\</p>\\<p>


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#127841 Całka funkcji trygonometrycznej 6

Napisane przez Mariusz M w 27.07.2016 - 18:58

Jarek podstawienie za tangens połowy kąta jest dobre ale ja proponowałbym złożenie podstawienia Eulera z podstawieniem za sinus

 

\sqrt{\sin^{2}{x}+2}=t-\sin{x}

 

Później podstawienie  u=t^2

aby mieć nieco mniej współczynników w rozkładzie na sumę ułamków prostych


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#127840 Całka funkcji trygonometrycznej 4

Napisane przez Mariusz M w 27.07.2016 - 18:28

Jarek mógłbyś pokazać jak tę całkę policzyć   Na oko wyjdzie eliptyczna

a zdaje się że ty takie całki liczyłeś

\int{\sqrt{sin{x}}\mbox{d}x}\\</p>\\<p>\sin{x}=t\\</p>\\<p>\mbox{d}t=\cos{x}\mbox{d}x\\</p>\\<p>\mbox{d}t=\sqrt{1-t^2}\mbox{d}x\\</p>\\<p>\int{\sqrt{t-t^3}\mbox{d}t}</p>\\<p>


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#127808 Całka niewymierna 4

Napisane przez Mariusz M w 23.07.2016 - 23:11

\int{\frac{\mbox{d}x}{\sqrt{x^3-x}}}\\</p>\\<p>=\int{\frac{\mbox{d}x}{\sqrt{x\left(x^2-1\right)}}}\\</p>\\<p>x=t^2\\</p>\\<p>\mbox{d}x=2t\mbox{d}t\\</p>\\<p>\int{\frac{2t\mbox{d}t}{\sqrt{t^2\left(t^4-1\right)}}}\\</p>\\<p>=2\int{\frac{\mbox{d}t}{\sqrt{-\left(1-t^2\right)\left(1+t^2\right)}}}\\</p>\\<p>t=\frac{1}{\sqrt{1-z^2}}\\</p>\\<p>t^2=\frac{1}{1-z^2}\\</p>\\<p>\frac{1}{t^2}=1-z^2</p>\\<p>\frac{1-t^2}{t^2}=-z^2\\</p>\\<p>\frac{1+t^2}{t^2}=2-z^2\\<br>\\1-t^2=-\frac{z^2}{1-z^2}\\</p>\\<p>1+t^2=-\frac{2-z^2}{1-z^2}\\</p>\\<p>\mbox{d}t=\frac{z}{\left(1-z^2\right)\sqrt{1-z^2}}\mbox{d}z\\</p>\\<p>2\int{\frac{1}{\sqrt{\frac{z^2}{1-z^2}\cdot\frac{2-z^2}{1-z^2}}}\cdot\frac{z}{\left(1-z^2\right)\sqrt{1-z^2}}\mbox{d}z}\\</p>\\<p>2\int{\frac{\mbox{d}z}{\sqrt{2-z^2}\sqrt{1-z^2}}}\\</p>\\<p>=\sqrt{2}\int{\frac{\mbox{d}z}{\sqrt{1-\frac{1}{2}z^2}\sqrt{1-z^2}}}\\</p>\\<p>=\sqrt{2}F\left(z,\frac{\sqrt{2}}{2}\right)+C\\</p>\\<p>=\sqrt{2}F\left(\frac{sqrt{t^2-1}}{t},\frac{\sqrt{2}}{2}\right)+C\\</p>\\<p>=\sqrt{2}F\left(\frac{\sqrt{x^2-x}}{x},\frac{\sqrt{2}}{2}\right)+C\\</p>\\<p>

 

F to całka eliptyczna

 

 

 

 


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#127799 Całka funkcji niewymiernej 7

Napisane przez Mariusz M w 23.07.2016 - 21:15

Spróbuj pierwszego lub drugiego podstawienia Eulera

 

\sqrt{x^2+2x+5}=t-x\\</p>\\<p>x^2+2x+5=t^2-2tx+x^2\\</p>\\<p>2x+5=t^2-2tx\\</p>\\<p>t^2-5=2tx+2x\\</p>\\<p>x\left(2t+2\right)=t^2-5\\</p>\\<p>x=\frac{t^2-5}{2t+2}\\</p>\\<p>x-3=\frac{t^2-6t-11}{2t+2}\\</p>\\<p>x-4=\frac{t^2-8t-13}{2t+2}\\</p>\\<p>t-x=\frac{2t^2+2t-t^2+5}{2t+2}=\frac{t^2+2t+5}{2t+2}\\</p>\\<p>\mbox{d}x=\frac{2t\left(2t+2\right)-2\left(t^2-5\right)}{\left(2t+2\right)^2}\mbox{d}t\\</p>\\<p>\mbox{d}x=\frac{2t^2+4t+10}{\left(2t+2\right)^2}\mbox{d}t\\</p>\\<p>\int{\frac{t^2+2t+5}{2t+2}\cdot\frac{2t+2}{t^2-6t-11}\cdot\frac{2t+2}{t^2-8t-13}\cdot\frac{2\left(t^2+2t+5\right)}{\left(2t+2\right)^2}\mbox{d}t}\\</p>\\<p>\int{\frac{\left(t^2+2t+5\right)^2}{\left(t^2-6t-11\right)\left(t^2-8t-13\right)\left(t+1\right)}\mbox{d}t}\\</p>\\<p>\int{\frac{\left(t^2+2t+5\right)^2}{\left(t-3-2\sqrt{5}\right)\left(t-3+2\sqrt{5}\right)\left(t-4-\sqrt{29}\right)\left(t-4+\sqrt{29}\right)\left(t+1\right)}\mbox{d}t}\\</p>\\<p>


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#127798 Całka niewymierna 2

Napisane przez Mariusz M w 23.07.2016 - 20:47

</p>\\<p></p>\\<p>\frac{1}{3}x^2-\frac{2}{3}x+\frac{5}{9}</p>\\<p>x^4-5x:3x^2+6x+7</p>\\<p>x^4+2x^3+\frac{7}{3}x^2</p>\\<p>-2x^3-\frac{7}{3}x^2-5x</p>\\<p>-2x^3-4x^2-\frac{14}{3}x</p>\\<p>\frac{5}{3}x^2-\frac{1}{3}x</p>\\<p>\frac{5}{3}x^2+\frac{10}{3}x+\frac{35}{9}</p>\\<p>-\frac{11}{3}x-\frac{35}{9}</p>\\<p>\int{\left(\frac{1}{3}x^2-\frac{2}{3}x+\frac{5}{9}\right)\mbox{d}x}-\frac{1}{9}\int{\frac{33x+35}{3x^2+6x+7}\mbox{d}x}\\</p>\\<p>=\frac{1}{9}x^3-\frac{1}{3}x^2+\frac{5}{9}x-\frac{1}{9}\int{\frac{33x+35}{3\left(x^2+2x+1\right)+4}\mbox{d}x}\\</p>\\<p>=\frac{1}{9}x^3-\frac{1}{3}x^2+\frac{5}{9}x-\frac{1}{9}\int{\frac{33x+35}{3\left(x+1\right)^2+4}\mbox{d}x}\\</p>\\<p>\sqrt{3}\left(x+1\right)=2t\\</p>\\<p>\sqrt{3}\mbox{d}x=2\mbox{d}t\\</p>\\<p>\mbox{d}x=\frac{2\sqrt{3}}{3}\mbox{d}t\\</p>\\<p>-\frac{2\sqrt{3}}{27}\int{\frac{22\sqrt{3}t+2}{4t^2+4}\mbox{d}t}\\</p>\\<p>-\frac{\sqrt{3}}{54}\int{\frac{22\sqrt{3}t+2}{t^2+1}\mbox{d}t}\\</p>\\<p>-\frac{\sqrt{3}}{54}\left(11\sqrt{3}\ln{\left|t^2+1\right|}+2\arctan{t}\right)+C\\</p>\\<p>-\frac{11}{18}\ln{\left|t^2+1\right|}-\frac{\sqrt{3}}{27}\arctan{t}+C\\</p>\\<p>-\frac{11}{18}\ln{\left|\frac{3x^2+6x+3+4}{4}\right|}-\frac{\sqrt{3}}{27}\arctan{\left(\frac{\sqrt{3}\left(x+1\right)}{2}\right)}+C\\</p>\\<p>-\frac{11}{18}\ln{\left|3x^2+6x+7\right|}-\frac{\sqrt{3}}{27}\arctan{\left(\frac{\sqrt{3}\left(x+1\right)}{2}\right)}+C\\</p>\\<p></p>\\<p>=\frac{1}{9}x^3-\frac{1}{3}x^2+\frac{5}{9}x-\frac{11}{18}\ln{\left|3x^2+6x+7\right|}-\frac{\sqrt{3}}{27}\arctan{\left(\frac{\sqrt{3}\left(x+1\right)}{2}\right)}+C\\</p>\\<p>


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#127635 Równanie różniczkowe Bernulliego

Napisane przez Mariusz M w 01.07.2016 - 18:45

\frac{\mbox{d}y}{\mbox{d}x}+xy=xy^3\\</p>\\<p>\frac{\mbox{d}y}{\mbox{d}x}=xy^3-xy\\</p>\\<p>\frac{\mbox{d}y}{\mbox{d}x}=x\left(y^3-y\right)\\</p>\\<p>\frac{\mbox{d}y}{y\left(y^2-1\right)}=x\mbox{d}x\\</p>\\<p>\frac{2\mbox{d}y}{y\left(y^2-1\right)}=2x\mbox{d}x\\</p>\\<p>\frac{2-2y^2+2y^2}{y\left(y^2-1\right)}\mbox{d}y=2x\mbox{d}x\\</p>\\<p>\left(-\frac{2}{y}+\frac{2y}{y^2-1}\right)\mbox{d}y=2x\mbox{d}x\\</p>\\<p>-2\ln{\left|y\right|}+\ln{\left|y^2-1\right|}=x^2+C\\</p>\\<p>\frac{y^2-1}{y^2}=Ce^{x^2}\\</p>\\<p>1-\frac{1}{y^2}=Ce^{x^2}\\</p>\\<p>1-Ce^{x^2}=\frac{1}{y^2}\\</p>\\<p>y^{2}=\frac{1}{1-Ce^{x^2}}\\</p>\\<p>


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