Textbook Question
58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:
b. Use substitution.
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.5.2b
2. Give an example of each of the following.
b. A repeated linear factor
Verified step by step guidance1
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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Taylor Polynomials
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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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
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
59. Two Methods
b. Evaluate ∫(x / √(x + 1)) dx using substitution.
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Textbook Question
75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:
s(t) = e⁻ᵗ sin t
c. Generalize part (b) and find the average value of the position on the interval [nπ, (n+1)π], for n = 0, 1, 2, ...
Textbook Question
82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.
b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.
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].)