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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.5.3
Capitolo 5, Problema 5.5.3

The composite function ƒ(g(𝓍)) consists of an inner function g and an outer function ƒ. If an integrand includes ƒ(g(𝓍)), which function is often a likely choice for a new variable u?

Guida verificata passo dopo passo
1
Identify the composite function given as ƒ(g(𝓍)), where g(𝓍) is the inner function and ƒ is the outer function.
Recall that when performing integration involving composite functions, substitution is a common technique to simplify the integral.
In substitution, we typically choose the inner function g(𝓍) as the new variable u because it simplifies the integrand and its differential du relates directly to dx.
Express the substitution as u = g(𝓍), then compute the differential du = g\' (𝓍) d𝓍 to replace parts of the integral accordingly.
Rewrite the integral entirely in terms of u and du, which often makes the integral easier to evaluate.

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Composite Functions

A composite function is formed when one function is applied to the result of another, written as ƒ(g(x)). Understanding how the inner function g(x) and outer function ƒ relate is essential for manipulating and simplifying expressions involving compositions.
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Evaluate Composite Functions - Special Cases

U-Substitution Method

U-substitution is a technique used in integration where a new variable u is chosen to simplify the integral. Typically, u is set equal to the inner function g(x) in a composite function to make the integral easier to evaluate.
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Substitution With an Extra Variable

Chain Rule in Integration

The chain rule relates the derivative of a composite function to the derivatives of its inner and outer functions. In integration, recognizing this structure helps identify the inner function as a candidate for substitution, reversing the chain rule process.
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Intro to the Chain Rule
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