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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.8.81a
Chapter 8, Problem 8.8.81a

81. Find the values of p for which each integral converges.
a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]

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1
Identify the integral and the parameter involved: we have the integral \( \int_1^2 \frac{dx}{x (\ln x)^p} \), and we want to find for which values of \( p \) this integral converges.
Check the behavior of the integrand near the limits of integration. Since the interval is from 1 to 2, and \( \ln 1 = 0 \), the potential issue is at the lower limit \( x = 1 \) where the denominator involves \( (\ln x)^p \) which tends to zero, possibly causing a singularity.
Make a substitution to analyze the behavior near \( x = 1 \): let \( t = \ln x \). Then \( dt = \frac{1}{x} dx \), so \( dx = x dt = e^t dt \). The integral limits change from \( x=1 \) to \( t=0 \), and from \( x=2 \) to \( t=\ln 2 \).
Rewrite the integral in terms of \( t \): \( \int_0^{\ln 2} \frac{e^t dt}{e^t t^p} = \int_0^{\ln 2} \frac{dt}{t^p} \). Now the integral simplifies to \( \int_0^{\ln 2} t^{-p} dt \).
Determine convergence of \( \int_0^{a} t^{-p} dt \) for some positive \( a \). This integral converges if and only if \( -p > -1 \), or equivalently \( p < 1 \). So the original integral converges for \( p < 1 \).

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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

An improper integral involves limits where the integrand becomes unbounded or the interval is infinite. To determine convergence, we analyze the behavior of the integrand near points causing potential issues, such as where the denominator approaches zero or infinity.
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Improper Integrals: Infinite Intervals

Behavior of the Natural Logarithm Function near 1

The natural logarithm ln(x) approaches 0 as x approaches 1 from the right. Since the integral's denominator includes (ln x)^p, understanding how ln(x) behaves near 1 is crucial to assess whether the integral converges or diverges.
Recommended video:
05:18
Derivative of the Natural Logarithmic Function

p-Test for Convergence of Integrals

The p-test helps determine convergence of integrals of the form ∫ (1 / t^p) dt near zero or infinity. By substituting t = ln(x), the integral can be analyzed using the p-test to find values of p for which the integral converges.
Recommended video:
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Choosing a Convergence Test
Related Practice
Textbook Question

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

a. Find the volume of the solid.

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Textbook Question

Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.

a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?

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Textbook Question

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

a. u = 1/(x + 1)

What is the value of the integral?

Textbook Question

Consider the region bounded by the graphs of

y = ln(x), y = 0, and x = e.

a. Find the area of the region.

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Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from -1 to 1 of (x² + 1) dx

Textbook Question

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.