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

Considérons une intégrale dont l’argument peut s’écrire comme la dérivée de la règle de chaîne d’une fonction composite F(g(x)). Pour résoudre l’intégrale, le processus consiste à inverser la différenciation de la règle de chaîne.Ici, une nouvelle variable, u, est définie comme g(x). Alors différenciez u par rapport à x. Cela peut être réarrangé en termes de du.Pour changer les limites, lorsque x=a, u égal à g(a) devient la nouvelle limite inférieure, et lorsque x=b, u égal à g(b) devient la nouvelle limite supérieure.L’intégrale originale est ensuite réécrite en substituant u et du, et en remplaçant respectivement les limites a et b par g(a) et g(b). En intégrant le nouvel intégrande par rapport à vous et en appliquant les limites changeantes, on obtient l’expression numérique finale.Un exemple de substitution se trouve en génie électrique, où elle sert à déterminer la charge totale qui passe dans un circuit sur un intervalle de temps donné. Ici, la charge totale est trouvée en calculant l’intégrale définie par rapport au temps. Comme le courant est donné par une fonction complexe, la substitution facilite la résolution de l’intégrale.

règle de substitution appliquée aux intégrales définies

Substitution Rule Applied to Definite Integrals

Lorsque vous calculez une int&#xE9;grale d&#xE9;finie dont l&#x2019;int&#xE9;grande correspond &#xE0; la structure d&#x2019;une fonction compos&#xE9;e, la m&#xE9;thode de substitution constitue un moyen efficace de simplifier le calcul. Cette m&#xE9;thode repose sur l&#x2019;inversion de la r&#xE8;gle de la cha&#xEE;ne lors de la d&#xE9;rivation, ce qui permet de r&#xE9;&#xE9;crire une expression complexe sous une forme plus simple. Lorsque l&#x2019;int&#xE9;grande contient une fonction interne accompagn&#xE9;e de sa d&#xE9;riv&#xE9;e, la substitution r&#xE9;duit naturellement la complexit&#xE9; du probl&#xE8;me.Le principe fondamental de la substitution pour les int&#xE9;grales d&#xE9;finies est r&#xE9;sum&#xE9; par la relation pr&#xE9;sent&#xE9;e ci-dessous.\begin{equation*}\int_a^b f(g(x))\,g&#x27;(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du\\end{equation*}La d&#xE9;marche commence par l&#x2019;identification de la fonction interne et par l&#x2019;introduction d&#x2019;une nouvelle variable destin&#xE9;e &#xE0; la repr&#xE9;senter. La d&#xE9;rivation de cette nouvelle variable fournit une diff&#xE9;rentielle correspondante, qui remplace la diff&#xE9;rentielle initiale. Parall&#xE8;lement, les bornes d&#x2019;int&#xE9;gration sont mises &#xE0; jour en &#xE9;valuant la fonction interne aux bornes initiales. Cela garantit que l&#x2019;int&#xE9;grale enti&#xE8;re est exprim&#xE9;e de mani&#xE8;re coh&#xE9;rente en termes de la nouvelle variable, sans qu&#x2019;il soit n&#xE9;cessaire de revenir &#xE0; la variable d&#x2019;origine apr&#xE8;s l&#x2019;int&#xE9;gration.Une fois la substitution r&#xE9;alis&#xE9;e, l&#x2019;int&#xE9;grale devient plus simple &#xE0; &#xE9;valuer, car l&#x2019;int&#xE9;grande ne d&#xE9;pend plus que d&#x2019;une seule variable. L&#x2019;application directe des nouvelles bornes donne alors la valeur num&#xE9;rique finale. Cette approche simplifie des calculs qui seraient autrement difficiles, voire impraticables, &#xE0; l&#x2019;aide des techniques d&#x2019;int&#xE9;gration usuelles.La substitution joue un r&#xF4;le cl&#xE9; dans de nombreux domaines appliqu&#xE9;s, notamment en &#xE9;lectrotechnique. Par exemple, la d&#xE9;termination de la charge &#xE9;lectrique totale qui circule dans un circuit au cours d&#x2019;un intervalle de temps donn&#xE9; n&#xE9;cessite l&#x2019;accumulation du courant sur cet intervalle. Lorsque le courant est d&#xE9;crit par une expression complexe, la substitution permet d&#x2019;&#xE9;valuer l&#x2019;int&#xE9;grale de mani&#xE8;re efficace, ce qui en fait un outil &#xE0; la fois pratique et puissant pour l&#x2019;analyse des syst&#xE8;mes r&#xE9;els.

Règle de substitution appliquée aux intégrales définies

Lorsque vous calculez une int&#xE9;grale d&#xE9;finie dont l&#x2019;int&#xE9;grande correspond &#xE0; la structure d&#x2019;une fonction compos&#xE9;e, la ...

substitution-rule-applied-to-definite-integrals

Education

JoVE Core

Calculus

Unit

Integrals

KEY TERMS AND CONCEPTS

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.

Using Substitution to Evaluate Definite Integrals | JoVE K12

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calculus

Using Substitution to Evaluate Definite Integrals

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