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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 8.5.2d
Chapter 8, Problem 8.5.2d

2. Give an example of each of the following.
d. A repeated irreducible quadratic factor

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1
Understand what an irreducible quadratic factor is: it is a quadratic polynomial (degree 2) that cannot be factored further over the real numbers, meaning it has no real roots.
Recognize that a repeated irreducible quadratic factor means the quadratic factor appears more than once in the factorization of a polynomial.
Construct an example by choosing an irreducible quadratic, such as \(x^{2} + 1\), which has no real roots and cannot be factored over the reals.
Form a polynomial where this quadratic factor is repeated, for example, by squaring it: \((x^{2} + 1)^{2}\).
Write the full polynomial explicitly: \((x^{2} + 1)^{2} = (x^{2} + 1)(x^{2} + 1)\), which clearly shows the repeated irreducible quadratic factor.

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Key Concepts

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

Irreducible Quadratic Factor

An irreducible quadratic factor is a quadratic polynomial that cannot be factored into real linear factors. Typically, it has a negative discriminant (b² - 4ac < 0), meaning it has no real roots. For example, x² + 1 is irreducible over the real numbers.
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Repeated Factor

A repeated factor in a polynomial is a factor that appears more than once, indicating multiplicity greater than one. For instance, (x - 2)² means the factor (x - 2) is repeated twice. Repeated factors affect the shape of the graph and the behavior of the function at the root.
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Partial Fraction Decomposition: Repeated Linear Factors

Combining Repeated and Irreducible Quadratic Factors

A repeated irreducible quadratic factor is a quadratic polynomial that is irreducible and appears with multiplicity greater than one in a polynomial. For example, (x² + 1)² is a repeated irreducible quadratic factor because x² + 1 cannot be factored further and is squared.
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Related Practice
Textbook Question

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

d. If ∫(from 1 to ∞) x^(-p) dx exists, then ∫(from 1 to ∞) x^(-q) dx exists (where q > p).

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Textbook Question

81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

d. Show that, in general, if n is odd, then Iₙ = -½ e⁻ˣ² pₙ₋₁(x), where pₙ₋₁ is a polynomial of degree n - 1.

Textbook Question

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

69. Let f(x) = sin(eˣ).

d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1.

Textbook Question

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

66. Let f(x) = cos(x²).

d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

Textbook Question

101. Many methods needed Show that the integral from ∫(from 0 to ∞)(sqrt(x) * ln x) / (1 + x)^2 dx equals pi, following these steps

d. Evaluate the remaining integral using the change of variables z = sqrt(x)

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Textbook Question

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

d. ∫(1/eˣ) dx = ln eˣ + C.