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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.5.3
Chapter 5, Problem 5.5.3

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

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1
Identify the composite function given as Ζ’(g(𝓍)), where g(𝓍) is the inner function and Ζ’ 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Composite Functions

A composite function is formed when one function is applied to the result of another, written as Ζ’(g(x)). Understanding how the inner function g(x) and outer function Ζ’ 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.
Recommended video:
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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.
Recommended video:
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Intro to the Chain Rule
Related Practice
Textbook Question

Areas of regions Find the area of the region bounded by the graph of Ζ’ and the 𝓍-axis on the given interval.


Ζ’(𝓍) = 𝓍³ ― 1 on [―1, 2]

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Textbook Question

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.

{Use of Tech} v = 4 √(t +1) (mi/hr) . for 0 ≀ t ≀ 15 ; n = 5     

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Textbook Question

Derivatives of integrals Simplify the following expressions.


d/dz βˆ«ΒΉβ°β‚›α΅’β‚™ β‚‚ dt /(t⁴ + 1)

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Textbook Question

Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.


Ζ’(𝓍) = 8 ― 2𝓍 on [0, 4]

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Textbook Question

Variations on the substitution method Evaluate the following integrals.                                                                                                        

                                                                                                                                                                    

 βˆ« 𝓍/(βˆ›π“ + 4) d𝓍

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Textbook Question

Evaluate βˆ«β‚ƒβΈ Ζ’ β€²(t) dt , where Ζ’ β€² is continuous on [3, 8], Ζ’(3) = 4, and Ζ’(8) = 20 .

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