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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.9.87d
Capitolo 8, Problema 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).

Guida verificata passo dopo passo
1
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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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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Pratica correlata
Domanda del libro di testo

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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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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Domanda del libro di testo

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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Domanda del libro di testo

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

68. Let f(x) = e^(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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77. Tabular integration Consider the integral ∫ f(x)g(x) dx, where f can be differentiated repeatedly and g can be integrated repeatedly

Let Gₖ represent the result of calculating k indefinite integrals of g (omitting constants of integration).

d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts.

Evaluate ∫ x² e^(x/2) dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in.

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82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.



e. Does this pattern continue? Is it true that A(1, ln b) = a² * A(a, (ln b)/a)?

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