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

Consideremos una integral cuyo argumento puede escribirse como la derivada de la regla de cadena de una función compuesta F(g(x)). Para resolver la integral, el proceso implica invertir la diferenciación de la regla de cadena.Aquí, una nueva variable, u, se define como g(x). Luego diferenciamos u respecto a x. Esto puede reorganizarse en términos de du.Para cambiar los límites, cuando x=a, u es igual a g(a) se convierte en el nuevo límite inferior, y cuando x=b, u es igual a g(b) se convierte en el nuevo límite superior.La integral original se reescribe sustituyendo u y du, y sustituyendo los límites a y b por g(a) y g(b), respectivamente. Integrar el nuevo integrando respecto a tu y aplicar los límites cambiantes da la expresión numérica final.Un ejemplo de sustitución se encuentra en ingeniería eléctrica, donde se utiliza para determinar la carga total que pasa por un circuito durante un intervalo de tiempo dado. Aquí la carga total se obtiene calculando la integral definida respecto al tiempo. Como la corriente está dada por una función compleja, la sustitución facilita la resolución de la integral.

regla de sustitución aplicada a integrales definidas

Al evaluar una integral definida cuyo integrando presenta la estructura de una funci&#xF3;n compuesta, el m&#xE9;todo de sustituci&#xF3;n ofrece una forma eficiente de simplificar el c&#xE1;lculo. Este m&#xE9;todo se basa en invertir la regla de la cadena de la diferenciaci&#xF3;n, lo que permite reescribir una expresi&#xF3;n compleja de forma m&#xE1;s simple. Cuando el integrando contiene una funci&#xF3;n interna y su derivada, la sustituci&#xF3;n reduce naturalmente la complejidad del problema.La idea central de la sustituci&#xF3;n para integrales definidas se resume en la relaci&#xF3;n que se muestra a continuaci&#xF3;n.\begin{equation*}\int_a^b f(g(x))\,g&#x27;(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du\\end{equation*}El proceso comienza identificando la funci&#xF3;n interna e introduciendo una nueva variable para representarla. Al diferenciar esta nueva variable, se obtiene el diferencial correspondiente que reemplaza al original. Al mismo tiempo, los l&#xED;mites de integraci&#xF3;n se actualizan evaluando la funci&#xF3;n interna en los extremos originales. Esto garantiza que toda la integral se exprese de forma consistente en t&#xE9;rminos de la nueva variable, eliminando la necesidad de volver a la variable original despu&#xE9;s de la integraci&#xF3;n.Una vez completada la sustituci&#xF3;n, la integral se vuelve m&#xE1;s sencilla de evaluar, ya que el integrando ahora depende de una sola variable. Al aplicar los l&#xED;mites actualizados, se obtiene directamente el valor num&#xE9;rico final. Este enfoque agiliza los c&#xE1;lculos que, de otro modo, ser&#xED;an dif&#xED;ciles o poco pr&#xE1;cticos utilizando t&#xE9;cnicas de integraci&#xF3;n est&#xE1;ndar.La sustituci&#xF3;n desempe&#xF1;a un papel crucial en diversos campos aplicados, incluida la ingenier&#xED;a el&#xE9;ctrica. Por ejemplo, determinar la carga el&#xE9;ctrica total que fluye a trav&#xE9;s de un circuito durante un intervalo de tiempo determinado requiere acumular la corriente durante ese intervalo. Cuando la corriente se describe con una expresi&#xF3;n compleja, la sustituci&#xF3;n permite evaluar la integral de forma eficiente, lo que la convierte en una herramienta pr&#xE1;ctica y potente para analizar sistemas del mundo real.

Regla de Sustitución Aplicada a Integrales Definidas

Al evaluar una integral definida cuyo integrando presenta la estructura de una funci&#xF3;n compuesta, el m&#xE9;todo de sustituci&#xF3;n ofrece una forma ...

substitution-rule-applied-to-definite-integrals

Education

Educación

JoVE Core

Calculus

Cálculo

Unit

Unidad

Integrals

Integrales

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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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