Textbook Question
Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x₁ and x₂ in I, x₂ ≠ x₁ implies f(x₂) ≠ f(x₁).
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 7.3.101
Evaluate the integrals in Exercises 97–110.
101. ∫ (log₁₀x / x) dx
Verified step by step guidance1
Recognize that the integral is \( \int \frac{\log_{10} x}{x} \, dx \). Since the logarithm is base 10, recall the change of base formula: \( \log_{10} x = \frac{\ln x}{\ln 10} \), where \( \ln x \) is the natural logarithm.
Rewrite the integral using the natural logarithm: \( \int \frac{\log_{10} x}{x} \, dx = \int \frac{\frac{\ln x}{\ln 10}}{x} \, dx = \frac{1}{\ln 10} \int \frac{\ln x}{x} \, dx \).
Focus on the integral \( \int \frac{\ln x}{x} \, dx \). To solve this, use substitution: let \( t = \ln x \), which implies \( dt = \frac{1}{x} dx \).
Rewrite the integral in terms of \( t \): \( \int t \, dt \), since \( \frac{\ln x}{x} dx = t \, dt \).
Integrate \( \int t \, dt \) to get \( \frac{t^2}{2} + C \), then substitute back \( t = \ln x \) to express the answer in terms of \( x \). Finally, multiply by the constant \( \frac{1}{\ln 10} \) from step 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Logarithmic Functions
Integrating logarithmic functions often involves recognizing the form and applying substitution or integration by parts. For example, integrals involving log(x) can be simplified by rewriting the logarithm or using properties of logarithms to facilitate integration.
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Graphs of Logarithmic Functions
Change of Logarithm Base
Logarithms with different bases can be converted using the formula log_a(x) = ln(x) / ln(a). This is useful because natural logarithms (ln) are easier to integrate and differentiate, allowing the integral to be expressed in terms of ln(x) for simpler calculation.
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05:36Change of Base Property
Substitution Method in Integration
The substitution method involves changing variables to simplify an integral. For integrals like ∫ (log₁₀x / x) dx, setting u = log₁₀x or u = ln(x) can transform the integral into a more manageable form, making it easier to evaluate.
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07:33Euler's Method
Related Practice
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)
Textbook Question
Suppose that the differentiable function y = f(x) has an inverse and that the graph of f passes through the point (2, 4) and has a slope of 1/3 there. Find the value of df⁻¹/dx at x = 4.
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11. Show that if positive functions f(x) and g(x) grow at the same rate as x→∞, then f=O(g) and g=O(f).
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Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
21. y=arccos(x²)
Textbook Question
84. Find lim(x→∞) (√(x² + 1) - √x).
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