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

Une intégrale avec une fonction non rationnelle est difficile à évaluer avec des méthodes standard.Considérons une tige dont la densité de masse linéaire est donnée en termes de densité linéaire constante, d’une longueur caractéristique et de la distance par rapport au côté gauche.L’objectif est de déterminer la masse de la tige, ce qui nécessite d’intégrer cette fonction de densité sur la longueur de la tige.Les racines du cube compliquent l’intégrale, donc une substitution rationalisante devient utile.Introduire une nouvelle variable u, définie comme u égal à la racine cubique de x, convertit l’expression en une forme rationnelle. À partir de là, x peut être pris comme le cube de u, et le différentiel dx en découle en conséquence. Les limites d’intégration sont ajustées pour correspondre à la nouvelle variable.En substituant ces expressions dans l’intégrale, on obtient une équation entièrement écrite en termes de u. Après avoir fait ces hypothèses, l’intégrale se simplifie en une forme polynomiale simple.Cette intégrale transformée est plus gérable, et la division longue polynomiale aide à simplifier la fonction rationnelle résultante.Après réécriture de l’expression en termes de u, évaluer l’intégrale avec les limites mises à jour donne la masse totale de la tige.De cette manière, l’intégrale est résolue en utilisant la substitution rationalisante.

substitution rationalisante

Rationalizing Substitutions

Les int&#xE9;grales impliquant des fonctions non rationnelles sont souvent difficiles &#xE0; calculer par des m&#xE9;thodes classiques, notamment lorsque l&#x2019;int&#xE9;grande contient des radicaux. La substitution rationalisante offre une m&#xE9;thode syst&#xE9;matique pour simplifier ces int&#xE9;grales en les convertissant en formes rationnelles plus faciles &#xE0; traiter.
Consid&#xE9;rez une tige dont la masse lin&#xE9;ique d&#xE9;pend d&#x2019;une densit&#xE9; lin&#xE9;ique constante, d&#x2019;une longueur caract&#xE9;ristique et de la distance &#xE0; l&#x2019;extr&#xE9;mit&#xE9; gauche de la tige. Pour d&#xE9;terminer la masse totale, il faut int&#xE9;grer cette densit&#xE9; le long de la tige. Lorsque la densit&#xE9; comprend des termes en racine cubique de la variable de position, l&#x2019;int&#xE9;gration directe devient fastidieuse, ce qui motive l&#x2019;emploi d&#x2019;une substitution rationalisante.
Pour simplifier l&#x2019;int&#xE9;grale, une nouvelle variable u est introduite et d&#xE9;finie par u = racine cubique de la variable d&#x2019;origine. Cette substitution permet d&#x2019;exprimer la variable initiale comme une simple puissance de u, et la diff&#xE9;rentielle est r&#xE9;&#xE9;crite en cons&#xE9;quence. Les bornes d&#x2019;int&#xE9;gration sont &#xE9;galement modifi&#xE9;es pour tenir compte du changement de variable. Ainsi, l&#x2019;int&#xE9;grande non rationnel est transform&#xE9; en une fonction rationnelle de u.
Apr&#xE8;s substitution, l&#x2019;int&#xE9;grale est enti&#xE8;rement exprim&#xE9;e en termes de la nouvelle variable. Sous les hypoth&#xE8;ses donn&#xE9;es concernant les constantes de la fonction de densit&#xE9;, l&#x2019;expression se simplifie encore, se r&#xE9;duisant souvent &#xE0; une fonction rationnelle pouvant &#xEA;tre trait&#xE9;e par des techniques alg&#xE9;briques comme la division euclidienne des polyn&#xF4;mes. Ce processus d&#xE9;compose l&#x2019;int&#xE9;grande en termes plus simples, ais&#xE9;s &#xE0; int&#xE9;grer.
L&#x2019;&#xE9;valuation de l&#x2019;int&#xE9;grale transform&#xE9;e, avec les bornes adapt&#xE9;es, permet d&#x2019;obtenir la masse totale de la tige. La substitution rationalisante constitue donc une strat&#xE9;gie efficace pour calculer des int&#xE9;grales faisant intervenir des radicaux, en les transformant en formes polynomiales ou rationnelles plus simples &#xE0; traiter.

Substitution rationalisante

Les int&#xE9;grales impliquant des fonctions non rationnelles sont souvent difficiles &#xE0; calculer par des m&#xE9;thodes classiques, notamment lorsque ...

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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

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Integration by Parts in AC Circuit Analysis

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integration by parts: definite integrals

Using Parts and Substitution for Area

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Solving Secant and Tangent Integrals

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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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Riemann sums estimate the area under a curve when an exact definite integral is hard to find. This often happens when a function has no known antiderivative or cannot be written in a closed form. It also happens when the function comes from measured data instead of a formula. The method works by splitting the interval of integration into equal parts. Each subinterval gets a rectangle, and the rectangle height comes from the function value at one chosen point in that part. Adding the rectangle...

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improper integrals: discontinuous integrands

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Improper Integrals at an Endpoint

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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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