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.
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 8.9.87d
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).
Verified step by step guidance1
Recall the integral in question is an improper integral of the form \(\int_1^{\infty} x^{-p} \, dx\). The convergence of this integral depends on the value of the exponent \(p\).
Determine the condition for convergence of \(\int_1^{\infty} x^{-p} \, dx\). This integral converges if and only if \(p > 1\). This is because the antiderivative is \(\frac{x^{-p+1}}{-p+1}\), and the limit as \(x \to \infty\) exists only when \(-p + 1 < 0\), or equivalently \(p > 1\).
Given that \(\int_1^{\infty} x^{-p} \, dx\) exists (converges), we know \(p > 1\). Now consider \(q > p\). Since \(q\) is greater than \(p\) and \(p > 1\), it follows that \(q > 1\) as well.
Because \(q > 1\), the integral \(\int_1^{\infty} x^{-q} \, dx\) also converges by the same reasoning as for \(p\). Thus, if the integral converges for \(p\), it must also converge for any \(q > p\).
Therefore, the statement is true: if \(\int_1^{\infty} x^{-p} \, dx\) exists, then \(\int_1^{\infty} x^{-q} \, dx\) exists for all \(q > p\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals and Convergence
Improper integrals extend the concept of definite integrals to infinite intervals or unbounded functions. Convergence means the integral approaches a finite value as the limit approaches infinity. Determining convergence often involves comparing the integrand's behavior at infinity.
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Improper Integrals: Infinite Intervals
p-Series Integral Test
The integral ∫₁^∞ x^(-p) dx converges if and only if p > 1. This is a fundamental result used to test convergence of integrals and series with power functions. It helps determine whether the area under the curve is finite over an infinite interval.
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04:30P-Series and Harmonic Series
Comparison of Exponents in Power Functions
For power functions x^(-p), a larger exponent p means the function decreases faster as x → ∞. If the integral converges for some p, it will also converge for any q > p because x^(-q) decays faster, ensuring the integral's convergence.
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Guided course
7:39Introduction to Exponent Rules
Related Practice
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
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).
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
2. Give an example of each of the following.
d. A repeated irreducible quadratic factor