Name: using substitution to evaluate definite integrals
Uploaded: 2026-01-12T12:24:59.000Z
Duration: 1 min 24 s
Channel: JoVE

Rozważmy całkę, której argument można zapisać jako pochodną reguły łańcuchowej funkcji złożonej F(g(x)). Aby rozwiązać całkę, proces polega na odwróceniu różniczkowania reguł łańcuchowych.Tutaj nowa zmienna, u, definiowana jest jako g(x). Następnie różniczkuj u względem x. Można to przeorganizować w kategoriach du.Aby zmienić granice, gdy x=a, u równa się g(a) staje się nową dolną granicą, a gdy x=b, u równa się g(b) staje się nową górną granicą.Oryginalna całka jest następnie przepisywana, podstawiając u i du oraz zastępując granice a i b odpowiednio g(a) i g(b). Całkowanie nowego całkownika względem U i zastosowanie zmieniających się granic daje ostateczne wyrażenie liczbowe.Przykładem podstawienia jest inżynieria elektryczna, gdzie służy do wyznaczania całkowitego ładunku przechodzącego przez obwód w określonym przedziale czasowym. Tutaj ładunek całkowity otrzymuje się przez obliczenie całki określonej względem czasu. Ponieważ prąd jest dany przez funkcję zespoloną, podstawienie ułatwia rozwiązanie całki.

reguła podstawiania zastosowana do całek oznaczonych

Podczas obliczania ca&#x142;ki oznaczonej, kt&#xF3;rej funkcja podca&#x142;kowa ma struktur&#x119; funkcji z&#x142;o&#x17C;onej, metoda podstawiania zapewnia efektywny spos&#xF3;b uproszczenia oblicze&#x144;. Metoda ta opiera si&#x119; na odwr&#xF3;ceniu regu&#x142;y &#x142;a&#x144;cuchowej stosowanej w r&#xF3;&#x17C;niczkowaniu, co pozwala na zapisanie skomplikowanego wyra&#x17C;enia w prostszej formie. Gdy funkcja podca&#x142;kowa zawiera funkcj&#x119; wewn&#x119;trzn&#x105; i jej pochodn&#x105;, podstawianie naturalnie zmniejsza z&#x142;o&#x17C;ono&#x15B;&#x107; problemu.G&#x142;&#xF3;wn&#x105; ide&#x119; podstawiania w ca&#x142;kach oznaczonych przedstawia poni&#x17C;sza zale&#x17C;no&#x15B;&#x107;.\begin{equation*}\int_a^b f(g(x))\,g&#x27;(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du\\end{equation*}Proces rozpoczyna si&#x119; od zidentyfikowania funkcji wewn&#x119;trznej i wprowadzenia nowej zmiennej reprezentuj&#x105;cej j&#x105;. R&#xF3;&#x17C;niczkowanie tej nowej zmiennej daje odpowiadaj&#x105;c&#x105; jej r&#xF3;&#x17C;niczk&#x119;, kt&#xF3;ra zast&#x119;puje pierwotn&#x105; r&#xF3;&#x17C;niczk&#x119;. Jednocze&#x15B;nie granice ca&#x142;ki s&#x105; aktualizowane poprzez obliczenie funkcji wewn&#x119;trznej na kra&#x144;cach pierwotnego przedzia&#x142;u. Gwarantuje to, &#x17C;e ca&#x142;a ca&#x142;ka jest wyra&#x17C;ona sp&#xF3;jnie wzgl&#x119;dem nowej zmiennej, eliminuj&#x105;c potrzeb&#x119; powrotu do pierwotnej zmiennej po wykonaniu ca&#x142;kowania.Po zako&#x144;czeniu podstawienia ca&#x142;ka oznaczona staje si&#x119; prostsza do obliczenia, poniewa&#x17C; funkcja podca&#x142;kowa zale&#x17C;y teraz od jednej zmiennej. Zastosowanie zaktualizowanych granic bezpo&#x15B;rednio daje ostateczn&#x105; warto&#x15B;&#x107; liczbow&#x105;. Takie podej&#x15B;cie usprawnia obliczenia, kt&#xF3;re w innym przypadku by&#x142;yby trudne lub niepraktyczne przy u&#x17C;yciu standardowych technik ca&#x142;kowania.Podstawianie odgrywa kluczow&#x105; rol&#x119; w r&#xF3;&#x17C;nych dziedzinach zastosowa&#x144;, w tym w in&#x17C;ynierii elektrycznej. Na przyk&#x142;ad, okre&#x15B;lenie ca&#x142;kowitego &#x142;adunku elektrycznego przep&#x142;ywaj&#x105;cego przez obw&#xF3;d w danym przedziale czasu wymaga akumulacji pr&#x105;du w tym przedziale czasu. Gdy pr&#x105;d jest opisany za pomoc&#x105; z&#x142;o&#x17C;onego wyra&#x17C;enia, podstawienie pozwala na efektywne obliczenie ca&#x142;ki, czyni&#x105;c je praktycznym i skutecznym narz&#x119;dziem do analizy system&#xF3;w rzeczywistych.

Reguła podstawiania zastosowana do całek oznaczonych

Podczas obliczania ca&#x142;ki oznaczonej, kt&#xF3;rej funkcja podca&#x142;kowa ma struktur&#x119; funkcji z&#x142;o&#x17C;onej, metoda podstawiania ...

substitution-rule-applied-to-definite-integrals

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Integrals

Całki

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

Substitution Rule Applied to Definite Integrals

using substitution to evaluate definite integrals

Substitution makes definite integrals easier to evaluate when the integrand is a composite function. It works by reversing the chain rule from differentiation. When the expression includes an inner function and its derivative, substitution can simplify the setup quickly.

The method starts by choosing the inner function and naming it with a new variable. Then its differential is found and used to replace the original variable terms. The limits of integration are also changed by plugging the original endpoints into the inner function. This keeps the integral written in one variable from start to finish.

After the change of variables, the integral usually becomes simpler to solve. Because the updated limits are used directly, there is no need to convert back to the original variable at the end. This makes the final numerical answer easier to obtain.

Substitution is useful when standard integration would be difficult or slow. It is also important in applied science and engineering. For example, in electrical engineering, it can help find the total electric charge that flows through a circuit over a time interval by integrating the current. If the current is given by a complicated expression, substitution makes the calculation more practical.

Using Substitution to Evaluate Definite Integrals

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substitution rule applied to definite integrals

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key terms and concepts

Learn how substitution simplifies definite integrals by changing variables, updating limits, and solving the integral directly. Includes a circuit current example.

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calculus

Using Substitution to Evaluate Definite Integrals

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