Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
a. lim(x→∞) tanh x
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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 7.3.143a
143.
a. Show that ∫ ln(x) dx = x ln(x) − x + C.
Verified step by step guidance1
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u = \ln(x)\) and \(dv = dx\). Then, compute \(du = \frac{1}{x} dx\) and \(v = x\).
Apply the integration by parts formula: \(\int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx\).
Simplify the integral: \(\int x \cdot \frac{1}{x} \, dx = \int 1 \, dx = x\).
Combine the results to write the integral as \(x \ln(x) - x + C\), where \(C\) is the constant of integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv − ∫v du. Choosing appropriate u and dv simplifies the integral, as in integrating ln(x) by setting u = ln(x) and dv = dx.
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Integration by Parts for Definite Integrals
Properties of the Natural Logarithm Function
The natural logarithm function, ln(x), is defined for x > 0 and has the derivative d/dx [ln(x)] = 1/x. Understanding its domain and derivative is essential when integrating ln(x), as it guides the choice of functions in integration by parts and ensures the integral is correctly evaluated.
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Constant of Integration
When computing indefinite integrals, the result includes an arbitrary constant C because differentiation of a constant is zero. This constant accounts for all possible antiderivatives, ensuring the general solution to the integral is complete.
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Related Practice
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + √x
Textbook Question
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
a. ln secθ + ln cosθ
Textbook Question
1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. x-3
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Textbook Question
10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)
Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
44. f(x) = 2x², x ≥ 0, a = 5