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.
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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 8.5.2d
2. Give an example of each of the following.
d. A repeated irreducible quadratic factor
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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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Partial Fraction Decomposition: Irreducible Quadratic Factors
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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11:26Partial 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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13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Related Practice
Textbook Question
57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The integral ∫ dx/(x² + 4x + 9) cannot be evaluated using a trigonometric substitution.
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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).
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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).
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Textbook Question
93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:
vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).
d. Which car ultimately gains the lead and remains in front?
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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. Using the substitution u = tan(x) in ∫ (tan²x / (tan x - 1)) dx leads to ∫ (u² / (u - 1)) du.
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