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

Consideriamo un integrale il cui argomento può essere scritto come derivata della regola della catena di una funzione composita F(g(x)). Per risolvere l'integrale, il processo comporta l'inversione della differenziazione delle regole della catena.Qui, una nuova variabile, u, è definita come g(x). Allora differenzia u rispetto a x. Questo può essere riorganizzato in termini di du.Per cambiare i limiti, quando x=a, u uguale a g(a) diventa il nuovo limite inferiore, e quando x=b, u uguale a g(b) diventa il nuovo limite superiore.L'integrale originale viene poi riscritto sostituendo u e du, e sostituendo i limiti a e b con g(a) e g(b), rispettivamente. Integrando il nuovo integrando rispetto a te e applicando i limiti che cambiano si ottiene l'espressione numerica finale.Un esempio di sostituzione si trova nell'ingegneria elettrica, dove viene usata per trovare la carica totale che passa attraverso un circuito in un dato intervallo di tempo. Qui la carica totale si trova calcolando l'integrale definito rispetto al tempo. Poiché la corrente è data da una funzione complessa, la sostituzione rende l'integrale più facile da risolvere.

regola di sostituzione applicata agli integrali definiti

Quando si valuta un integrale definito il cui integrando presenta la struttura di una funzione composta, il metodo di sostituzione offre un modo efficace per semplificare il calcolo. Questo metodo si basa sull&#x2019;applicazione inversa della regola della catena, consentendo di riscrivere un&#x2019;espressione complessa in una forma pi&#xF9; semplice. Quando l&#x2019;integrando contiene una funzione interna e la sua derivata, la sostituzione riduce naturalmente la complessit&#xE0; del problema.L&#x2019;idea centrale della sostituzione negli integrali definiti &#xE8; riassunta dalla relazione mostrata di seguito.\begin{equation*}\int_a^b f(g(x))\,g&#x27;(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du\\end{equation*}Il procedimento inizia individuando la funzione interna e introducendo una nuova variabile per rappresentarla. Derivando questa nuova variabile si ottiene il differenziale corrispondente, che sostituisce quello originario. Al contempo, i limiti di integrazione vengono aggiornati valutando la funzione interna agli estremi originari. Ci&#xF2; garantisce che l&#x2019;intero integrale sia espresso in modo coerente in funzione della nuova variabile, eliminando la necessit&#xE0; di tornare alla variabile originale dopo l&#x2019;integrazione.Una volta completata la sostituzione, l&#x2019;integrale risulta pi&#xF9; semplice da valutare perch&#xE9; l&#x2019;integrando dipende ora da una sola variabile. L&#x2019;applicazione diretta dei limiti aggiornati fornisce il valore numerico finale. Questo approccio snellisce calcoli che, altrimenti, sarebbero difficili o poco pratici con le tecniche standard di integrazione.La sostituzione svolge un ruolo fondamentale in diversi ambiti applicativi, tra cui l&#x2019;ingegneria elettrica. Per esempio, determinare la carica elettrica totale che attraversa un circuito in un dato intervallo di tempo richiede di accumulare la corrente lungo tale intervallo. Quando la corrente &#xE8; descritta da un&#x2019;espressione complessa, la sostituzione consente di valutare l&#x2019;integrale in modo efficiente, rendendola uno strumento pratico e potente per l&#x2019;analisi di sistemi reali.

Regola di sostituzione applicata agli integrali definiti

Quando si valuta un integrale definito il cui integrando presenta la struttura di una funzione composta, il metodo di sostituzione offre un modo efficace ...

substitution-rule-applied-to-definite-integrals

Education

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Calculus

Analisi matematica

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Integrals

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KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

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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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