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

Considere uma integral cujo argumento pode ser escrito como a derivada da regra da cadeia de uma função composta F(g(x)). Para resolver a integral, o processo envolve reverter a diferenciação da regra da cadeia.Aqui, uma nova variável, u, é definida como g(x). Então diferencie u em relação a x. Isso pode ser rearranjado em termos de du.Para alterar os limites, quando x=a, u igual a g(a) torna-se o novo limite inferior, e quando x=b, u igual a g(b) torna-se o novo limite superior.A integral original é então reescrita substituindo u e du, e substituindo os limites a e b por g(a) e g(b), respectivamente. Integrar o novo integrando em relação a você e aplicar os limites que mudam fornece a expressão numérica final.Um exemplo de substituição é encontrado em engenharia elétrica, onde ela é usada para encontrar a carga total que passa por um circuito em um determinado intervalo de tempo. Aqui, a carga total é encontrada calculando a integral definida em relação ao tempo. Como a corrente é dada por uma função complexa, a substituição facilita a resolução da integral.

regra de substituição aplicada a integrais definidas

Ao avaliar uma integral definida cujo integrando corresponde &#xE0; estrutura de uma fun&#xE7;&#xE3;o composta, o m&#xE9;todo de substitui&#xE7;&#xE3;o oferece uma maneira eficiente de simplificar o c&#xE1;lculo. Este m&#xE9;todo baseia-se na invers&#xE3;o da regra da cadeia da deriva&#xE7;&#xE3;o, permitindo que uma express&#xE3;o complexa seja reescrita em uma forma mais simples. Quando o integrando cont&#xE9;m uma fun&#xE7;&#xE3;o interna e sua derivada, a substitui&#xE7;&#xE3;o naturalmente reduz a complexidade do problema.A ideia central da substitui&#xE7;&#xE3;o para integrais definidas &#xE9; resumida pela rela&#xE7;&#xE3;o mostrada abaixo.\begin{equation*}\int_a^b f(g(x))\,g&#x27;(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du\\end{equation*}O processo inicia-se com a identifica&#xE7;&#xE3;o da fun&#xE7;&#xE3;o interna e a introdu&#xE7;&#xE3;o de uma nova vari&#xE1;vel para represent&#xE1;-la. Ao diferenciar essa nova vari&#xE1;vel, obt&#xE9;m-se um diferencial correspondente que substitui o original. Ao mesmo tempo, os limites de integra&#xE7;&#xE3;o s&#xE3;o atualizados avaliando-se a fun&#xE7;&#xE3;o interna nos limites originais. Isso garante que toda a integral seja expressa de forma consistente em termos da nova vari&#xE1;vel, eliminando a necessidade de voltar &#xE0; vari&#xE1;vel original ap&#xF3;s a integra&#xE7;&#xE3;o.Uma vez conclu&#xED;da a substitui&#xE7;&#xE3;o, a integral torna-se mais simples de avaliar, pois o integrando agora depende de uma &#xFA;nica vari&#xE1;vel. A aplica&#xE7;&#xE3;o direta dos limites atualizados fornece o valor num&#xE9;rico final. Essa abordagem simplifica c&#xE1;lculos que seriam dif&#xED;ceis ou impratic&#xE1;veis usando t&#xE9;cnicas padr&#xE3;o de integra&#xE7;&#xE3;o.A substitui&#xE7;&#xE3;o desempenha um papel fundamental em diversas &#xE1;reas aplicadas, incluindo a engenharia el&#xE9;trica. Por exemplo, determinar a carga el&#xE9;trica total que flui por um circuito em um determinado intervalo de tempo requer acumular a corrente nesse intervalo. Quando a corrente &#xE9; descrita por uma express&#xE3;o complexa, a substitui&#xE7;&#xE3;o permite que a integral seja avaliada de forma eficiente, tornando-se uma ferramenta pr&#xE1;tica e poderosa para analisar sistemas do mundo real.

Regra de Substituição Aplicada a Integrais Definidas

Ao avaliar uma integral definida cujo integrando corresponde &#xE0; estrutura de uma fun&#xE7;&#xE3;o composta, o m&#xE9;todo de substitui&#xE7;&#xE3;o ...

substitution-rule-applied-to-definite-integrals

Education

JoVE Core

Calculus

Cálculo

Unit

Unidade

Integrals

Integrais

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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

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...

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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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