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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 8.9.36
Chapter 8, Problem 8.9.36

7–58. Improper integrals Evaluate the following integrals or state that they diverge.
36. ∫ (from e² to ∞) 1/(x lnᵖ x) dx, p > 1

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Identify the integral to evaluate: \(\displaystyle \int_{e^{2}}^{\infty} \frac{1}{x (\ln x)^{p}} \, dx\) where \(p > 1\).
Recognize that this is an improper integral because the upper limit of integration is infinite. We rewrite it as a limit: \(\displaystyle \lim_{t \to \infty} \int_{e^{2}}^{t} \frac{1}{x (\ln x)^{p}} \, dx\).
Use the substitution \(u = \ln x\), which implies \(du = \frac{1}{x} dx\). This transforms the integral into \(\displaystyle \lim_{t \to \infty} \int_{\ln e^{2}}^{\ln t} \frac{1}{u^{p}} \, du\).
Simplify the limits of integration: since \(\ln e^{2} = 2\), the integral becomes \(\displaystyle \lim_{t \to \infty} \int_{2}^{\ln t} u^{-p} \, du\).
Evaluate the integral \(\int u^{-p} \, du\) for \(p \neq 1\), which is \(\frac{u^{-p+1}}{-p+1} + C\). Then apply the limits from 2 to \(\ln t\) and analyze the limit as \(t \to \infty\) to determine if the integral converges or diverges.

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Key Concepts

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

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, limits are used to define the integral as a limit of definite integrals over finite intervals. Determining convergence or divergence is essential for these integrals.
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Convergence Tests for Improper Integrals

To determine if an improper integral converges, comparison tests or p-integral tests are applied. For integrals of the form ∫ 1/(x (ln x)^p) dx from a finite number to infinity, the behavior of the integrand as x approaches infinity dictates convergence, often depending on the exponent p.
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Logarithmic Functions and Their Properties

Understanding the natural logarithm function ln(x) and its growth rate is crucial. Since ln(x) grows slower than any power of x, integrals involving ln(x) in the denominator require careful analysis of how powers of ln(x) affect convergence, especially in improper integrals extending to infinity.
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Properties of Functions
Related Practice
Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

45. ∫ sec²x tan¹ᐟ²x dx

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23-64. Integration Evaluate the following integrals.

50. ∫ 8(x² + 4)/[x(x² + 8)] dx

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Textbook Question

7–84. Evaluate the following integrals.

82. ∫ 1/(1 + tanx) dx

Textbook Question

48. Integral of sec³x Use integration by parts to show that:

∫ sec³x dx = (1/2) secx tanx + (1/2) ∫ secx dx

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95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.

96. ∫(from 0 to ∞) (sin²x)/x² dx = π/2

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Textbook Question

66. Integrating derivatives

Use integration by parts to show that if f' is continuous on [a, b], then

∫[a to b] f(x)f'(x) dx = (1/2)[f(b)² - f(a)²]

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