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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.5.2b
Chapter 8, Problem 8.5.2b

2. Give an example of each of the following.
b. A repeated linear factor

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1
Understand that a repeated linear factor in a polynomial is a linear factor (of the form \( (x - a) \)) that appears more than once, meaning it is raised to a power greater than 1.
Recall that a linear factor looks like \( (x - a) \), where \( a \) is a constant.
To create a repeated linear factor, take a linear factor and raise it to a power greater than 1, for example, \( (x - 3)^2 \).
An example polynomial with a repeated linear factor could be \( (x - 3)^2 (x + 1) \), where \( (x - 3) \) is the repeated linear factor.
This polynomial shows the repeated linear factor clearly, as \( (x - 3) \) appears twice, indicating multiplicity 2.

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Key Concepts

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

Polynomial Factors

Polynomial factors are expressions that multiply together to form a polynomial. Understanding how to factor polynomials into linear and nonlinear components is essential for simplifying and solving polynomial equations.
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Linear Factors

A linear factor is a polynomial of degree one, typically in the form (x - a), where 'a' is a root of the polynomial. Recognizing linear factors helps in breaking down polynomials into simpler parts.
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Linearization

Repeated Factors

A repeated factor occurs when a factor appears more than once in the factorization of a polynomial, such as (x - a)^n with n > 1. Identifying repeated factors is important for understanding multiplicity of roots and their effects on the graph.
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Related Practice
Textbook Question

49. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample:

c. ∫ v du = u·v - ∫ u dv

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Textbook Question

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.

The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]

(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project ""The exponential Eiffel Tower"")

b. With a = 0 and c = 2, find the equations of the lines tangent to both curves at x = 0

Textbook Question

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.

b. Find the area of the region bounded by the graph of g and the x-axis on the interval [0,2].

Textbook Question

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

45. ∫(0 to 1) e^(2x) dx; n = 25

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

Textbook Question

59. Two Methods

b. Evaluate ∫(x / √(x + 1)) dx using substitution.

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

71. Let f(x) = √(sin x).

b. Find an upper bound on the absolute error in the estimate from part (a) using Theorem 8.1. (Hint: |f''''(x)| ≤ 1 on [1,2].)