Textbook Question
7. Order the following functions from slowest growing to fastest growing as x→∞.
a. e^x
b. x^x
c. (ln x)^x
d. e^(x/2)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.2.47
Evaluate the integrals in Exercises 39–56.
47. ∫(from 2 to 4)dx/(x(ln x)²)
Verified step by step guidance1
Identify the integral to be evaluated: \(\int_{2}^{4} \frac{dx}{x (\ln x)^2}\).
Recognize that the integrand involves \(x\) and \(\ln x\), suggesting a substitution related to \(\ln x\).
Let \(u = \ln x\). Then, compute \(du\): since \(u = \ln x\), we have \(du = \frac{1}{x} dx\), which implies \(dx = x \, du\).
Rewrite the integral in terms of \(u\): substitute \(dx\) and \(x\) accordingly. The integral becomes \(\int_{u=\ln 2}^{u=\ln 4} \frac{x \, du}{x u^2} = \int_{\ln 2}^{\ln 4} \frac{du}{u^2}\).
Now, integrate \(\int \frac{1}{u^2} du = \int u^{-2} du\). Use the power rule for integration to find the antiderivative, then apply the limits \(u=\ln 2\) to \(u=\ln 4\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Rational Functions
This involves integrating functions expressed as ratios of polynomials or other expressions. Recognizing the form helps in choosing appropriate techniques such as substitution or partial fractions to simplify the integral.
Recommended video:
6:04
Intro to Rational Functions
Substitution Method
A technique where a part of the integral is replaced with a new variable to simplify the integral. For example, substituting u = ln(x) transforms the integral into a more manageable form involving u.
Recommended video:
07:33Euler's Method
Properties of the Natural Logarithm
Understanding the behavior and derivatives of ln(x) is crucial. Since d/dx(ln x) = 1/x, this relationship often guides substitution choices and helps simplify integrals involving ln(x).
Recommended video:
05:36Change of Base Property
Related Practice
Textbook Question
Evaluate the integrals in Exercises 53–76.
61. ∫(from 0 to 2)dt/√(8+2t²)
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Textbook Question
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))
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Textbook Question
Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.
sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞
cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1
tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1
sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1
csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1
coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1
Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.
63. tanh⁻¹(-1/2)
Textbook Question
In Exercises 139–142, find the length of each curve.
139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.
Textbook Question
Evaluate the integrals in Exercises 97–110.
107. ∫₀⁹ (2 log₁₀(x + 1) / (x + 1)) dx
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