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temata:17a-matematicka_analyza:integraly_jedne_promenne:main [2011/02/28 23:28] vagabund |
temata:17a-matematicka_analyza:integraly_jedne_promenne:main [2011/03/15 14:54] (aktuální) vagabund |
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|---|---|---|---|
| Řádek 94: | Řádek 94: | ||
| </box> | </box> | ||
| + | === Příklady === | ||
| + | |||
| + | <box round 90% green|**Příklady**> | ||
| + | |||
| + | Odvoďte rekurentní vzorec pro výpočet integrálu | ||
| + | |||
| + | <m>I_n(x) = \int{}{}{dx/{(1+x^2)^n}}, n \in N</m> | ||
| + | |||
| + | 1.) integrujeme pomocí per partes | ||
| + | |||
| + | <m>\int{}{}{dx/{(1+x^2)^n}} = x/{(1+x^2)^n} - \int{}{}{x.(-n((1+x^2)^{-n - 1}).2x)} = x/{(1+x^2)^n} + 2n\int{}{}{{x^2}/{(1+x^2)^{n + 1}}}</m> | ||
| + | |||
| + | Z toho | ||
| + | |||
| + | <m>\int{}{}{{x^2}/{(1+x^2)^{n + 1}}} = \int{}{}{{x^2 + 1 - 1}/{(1+x^2)^{n + 1}}} = \int{}{}{{x^2 + 1}/{(1+x^2)^{n + 1}}} - \int{}{}{{1}/{(1+x^2)^{n + 1}}} = \int{}{}{{1}/{(1+x^2)^n}} - \int{}{}{{1}/{(1+x^2)^{n + 1}}}</m> | ||
| + | |||
| + | Dáme dohromady | ||
| + | |||
| + | <m>\int{}{}{dx/{(1+x^2)^n}} = x/{(1+x^2)^n} + 2n(\int{}{}{{1}/{(1+x^2)^n}} - \int{}{}{{1}/{(1+x^2)^{n + 1}}})</m> | ||
| + | |||
| + | <m>I_n(x) = x/{(1+x^2)^n} + 2n(I_n(x+1) - I_{n + 1}(x))</m> | ||
| + | |||
| + | <m>I_n(x) = x/{(1+x^2)^n} + 2nI_n(x+1) - 2nI_{n + 1}(x)</m> | ||
| + | |||
| + | <m>2nI_{n + 1}(x) = x/{(1+x^2)^n} + 2nI_n(x+1) - I_n(x)</m> | ||
| + | |||
| + | <m>2nI_{n + 1}(x) = x/{(1+x^2)^n} + (2n- 1)I_n(x+1)</m> | ||
| + | |||
| + | <m>I_{n + 1}(x) = {x/{(1+x^2)^n} + (2n- 1)I_n(x+1)}/{2n}</m> | ||
| + | |||
| + | <m>I_{n + 1}(x) = x/{2n.(1+x^2)^n} + {(2n- 1)}/{2n}I_n(x+1)}</m> | ||
| + | |||
| + | 2.) spočítáma integrál pro n = 1 | ||
| + | |||
| + | <m>I_1(x) = \int{}{}{dx/{(1+x^2)}} = arctg(x)</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | <box round 90% green|**Příklady**> | ||
| + | |||
| + | <m>\int{}{}{{x^3 + 1}/{x^3 - x^2}dx}</m> | ||
| + | |||
| + | <m>{x^3 + 1}:{x^3 - x^2} = 1 + {x^2 + 1}/(x^3 - x^2)</m> | ||
| + | |||
| + | <m>{x^2 + 1}/{x^3 - x^2} = {x^2 + 1}/{x^2(x - 1)} = A/x + B/x^2 + C/{x - 1}</m> | ||
| + | |||
| + | <m>x^2 + 1 = Ax(x - 1) + B(x - 1) + Cx^2 = A(x^2 - x) + B(x - 1) + Cx^2</m> | ||
| + | |||
| + | <m>x^2: 1 = A + C</m> | ||
| + | |||
| + | <m>x: 0 = -A + B</m> | ||
| + | |||
| + | <m>1: 1 = -B</m> | ||
| + | |||
| + | <m>B = -1, A = -1, C = 2</m> | ||
| + | |||
| + | <m>{x^2 + 1}/{x^3 - x^2} = -1/x - 1/x^2 + 2/{x - 1}</m> | ||
| + | |||
| + | takže celkem | ||
| + | |||
| + | <m>\int{}{}{{x^3 + 1}/{x^3 - x^2}dx} = \int{}{}{1 - 1/x - 1/x^2 + 2/{x - 1}} = x - ln|x| + 1/x + 2ln|x - 1|</m> | ||
| + | |||
| + | |||
| + | |||
| + | </box> | ||
| === Teorie === | === Teorie === | ||
| Řádek 157: | Řádek 222: | ||
| </box> | </box> | ||
| + | |||
| + | === Numerické metody === | ||
| + | |||
| + | <box round 90% green|**Předpoklady**> | ||
| + | |||
| + | <a, b> ... interval | ||
| + | |||
| + | <m> h = {b - a}/n</m> | ||
| + | |||
| + | n ... počet částečných intervalů | ||
| + | |||
| + | <m>D_i = max \lbrace |f^{(i)}(x)|: x \in <a, b> \rbrace, i \in \lbrace 1, 2, 4\rbrace</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | <box round 90% green|**Obdélníková metoda**> | ||
| + | |||
| + | <m>\int{a}{b}{f(x)dx} \approx hy_0 + hy_1 + ... + hy_{n - 1} = h(y_0 + y_1 + ... + y_{n - 1})</m> | ||
| + | |||
| + | chyba metody: | ||
| + | |||
| + | <m>R_n <= {(b - a)^2 D_1}/n</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | <box round 90% green|**Lichoběžníková metoda**> | ||
| + | |||
| + | <m>\int{a}{b}{f(x)dx} \approx h{y_0 + y_1}/2 + h{y_1 + y_2}/2 + ... + h{y_{n-1} + y_n}/2 = {(b - a)}/{2n} (y_0 + 2y_1 + ... + 2y_{n - 1} + y_n)</m> | ||
| + | |||
| + | chyba metody: | ||
| + | |||
| + | <m>R_n <= {(b - a)^3 D_2}/{12n^2}</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | <box round 90% green|**Simpsonova metoda**> | ||
| + | |||
| + | <m>\int{a}{b}{f(x)dx} \approx {(b - a)}/{3n} (y_0 + 4y_1 + 2y_2 + ... + 2y_{n - 2} + 4y_{n - 1} + y_n)</m> | ||
| + | |||
| + | chyba metody: | ||
| + | |||
| + | <m>R_n <= {(b - a)^5 D_4}/{180n^4}</m> | ||
| + | |||
| + | Pozn. n musí být sudé | ||
| + | |||
| + | </box> | ||
| + | |||
| === Odkazy === | === Odkazy === | ||
| * http://cs.wikipedia.org/wiki/Integrál | * http://cs.wikipedia.org/wiki/Integrál | ||
| + | * http://www.wolframalpha.com/input/?i=int+1%2Fsqrt%281+%2B+x%5E2%29+dx | ||
| + | * http://mathonline.fme.vutbr.cz/Matematika-I/sc-5-sr-1-a-4/default.aspx | ||
| + | * | ||
| + | ===== Potvrzení ===== | ||
| + | |||
| + | <doodle single login|XX> | ||
| + | ^ OK ^ !!! ^ | ||
| + | </doodle> | ||
| + | |||
| + | {{tag>vagabund integraly per_partes parcialni_zlomky urcity_integral}} | ||
| + | |||
