Name: integrating radicals with rational substitution
Uploaded: 2026-01-12T12:24:35.000Z
Duration: 1 min 29 s
Channel: JoVE

Una integral con función no racional es difícil de evaluar usando métodos estándar.Consideremos una varilla cuya densidad de masa lineal se da en términos de densidad lineal constante, una longitud característica y la distancia desde el lado izquierdo.El objetivo es encontrar la masa de la varilla, lo que requiere integrar esta función de densidad a lo largo de la longitud de la varilla.Las raíces cúbicas complican la integral, por lo que una sustitución racionalizadora resulta útil.Introducir una nueva variable u, definida como u igual a la raíz cúbica de x, convierte la expresión en una forma racional. De esto, x puede tomarse como el cubo de u, y el diferencial dx se deduce en consecuencia. Los límites de integración se ajustan para ajustarse a la nueva variable.Sustituyendo estas expresiones en la integral se obtiene una ecuación escrita enteramente en términos de u. Tras hacer las suposiciones, la integral se simplifica a una forma polinómica simple.Esta integral transformada es más manejable, y la división larga polinómica ayuda a simplificar la función racional resultante.Tras reescribir la expresión en términos de u, evaluar la integral con los límites actualizados da la masa total de la varilla.De este modo, la integral se resuelve mediante la sustitución racionalizante.

sustituciones racionalizadoras

Rationalizing Substitutions

Las integrales que involucran funciones no racionales suelen ser dif&#xED;ciles de evaluar con t&#xE9;cnicas est&#xE1;ndar, especialmente cuando aparecen radicales en el integrando. La sustituci&#xF3;n racionalizadora proporciona un m&#xE9;todo sistem&#xE1;tico para simplificar dichas integrales, convirti&#xE9;ndolas en formas racionales m&#xE1;s f&#xE1;ciles de manejar.
Considere una varilla cuya densidad de masa lineal depende de una densidad lineal constante, una longitud caracter&#xED;stica y la distancia desde el extremo izquierdo de la varilla. Para determinar la masa total, es necesario integrar esta funci&#xF3;n de densidad sobre la longitud de la varilla. Cuando la densidad incluye t&#xE9;rminos de ra&#xED;z c&#xFA;bica de la variable de posici&#xF3;n, la integraci&#xF3;n directa se vuelve engorrosa, lo que justifica el uso de una sustituci&#xF3;n racionalizadora.
Para simplificar la integral, se introduce una nueva variable u, definida como la ra&#xED;z c&#xFA;bica de la variable original. Esta sustituci&#xF3;n permite expresar la variable original como una potencia simple de u, y el diferencial se reescribe en consecuencia. Los l&#xED;mites de integraci&#xF3;n tambi&#xE9;n se ajustan para reflejar el cambio de variables. Como resultado, el integrando no racional original se transforma en una funci&#xF3;n racional de u.
Tras la sustituci&#xF3;n, la integral se escribe completamente en t&#xE9;rminos de la nueva variable. Bajo los supuestos dados para las constantes en la funci&#xF3;n de densidad, la expresi&#xF3;n se simplifica a&#xFA;n m&#xE1;s, reduci&#xE9;ndose a menudo a una funci&#xF3;n racional que puede manejarse mediante t&#xE9;cnicas algebraicas como la divisi&#xF3;n de polinomios. Este proceso separa el integrando en t&#xE9;rminos m&#xE1;s simples y f&#xE1;ciles de integrar.
La evaluaci&#xF3;n de la integral transformada con los l&#xED;mites actualizados proporciona la masa total de la varilla. Por lo tanto, la sustituci&#xF3;n racionalizadora proporciona una estrategia eficaz para resolver integrales con radicales, convirti&#xE9;ndolas en formas polin&#xF3;micas o racionales m&#xE1;s manejables.

Sustituciones racionalizadoras

Las integrales que involucran funciones no racionales suelen ser dif&#xED;ciles de evaluar con t&#xE9;cnicas est&#xE1;ndar, especialmente cuando aparecen ...

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Techniques of Integration

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integrating radicals with rational substitution

Integrating radicals can be easier when a rationalizing substitution turns the expression into a rational form. This method is useful when standard techniques do not work well because the integrand contains a radical, such as a cube root.

The transcript uses a rod as a real-world example. Its linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Finding the total mass means integrating that density over the length of the rod. When the density includes cube root terms in the position variable, the direct integral becomes difficult to evaluate.

A rationalizing substitution fixes that problem by introducing a new variable, u, equal to the cube root of the original variable. The original variable is then written as a power of u, and the differential is rewritten as well. The limits of integration are also changed to match the new variable. After this step, the non-rational integrand becomes a rational function of u.

The transformed integral is then written entirely in terms of u. Under the stated assumptions for the constants in the density function, it can simplify further into a rational expression. In many cases, algebraic tools such as polynomial long division make the result easier to integrate. Evaluating the new integral with the updated limits gives the total mass of the rod.

Integrating Radicals with Rational Substitution

integrating-radicals-with-rational-substitution

integration by parts: indefinite integrals

integration-by-parts-indefinite-integrals

Integration by Parts in AC Circuit Analysis

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Using Parts and Substitution for Area

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Rational functions are ratios of two polynomials, and their integrals can be simplified by breaking them into smaller parts. A rational function is proper when the degree of the numerator is less than the degree of the denominator. When that is true, partial fraction decomposition rewrites the expression as a sum of simpler rational terms. To decompose a proper rational function, unknown constants are placed into the simpler terms. The denominator is then multiplied through to remove the...

rationalizing substitutions

Integrating radicals can be easier when a rationalizing substitution turns the expression into a rational form. This method is useful when standard techniques do not work well because the integrand contains a radical, such as a cube root. The transcript uses a rod as a real-world example. Its linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Finding the total mass means integrating that density over the length of...

approximate integration

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key terms and concepts

Learn how rationalizing substitution turns radical integrals into rational forms, using a rod mass example and updated limits of integration.

Integrating Radicals with Rational Substitution | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Integrating Radicals with Rational Substitution

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