Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
27. y = (1 - θ)tanh⁻¹(θ)
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.109
Evaluate the integrals in Exercises 97–110.
109. ∫ (dx / (x log₁₀x))
Verified step by step guidance1
Recognize that the integral involves the expression \(\frac{1}{x \log_{10} x}\). Since the logarithm is base 10, recall the change of base formula: \(\log_{10} x = \frac{\ln x}{\ln 10}\), where \(\ln\) is the natural logarithm.
Rewrite the integral using the change of base formula: \(\int \frac{dx}{x \log_{10} x} = \int \frac{dx}{x \frac{\ln x}{\ln 10}} = \int \frac{\ln 10}{x \ln x} \, dx\).
Factor out the constant \(\ln 10\) from the integral: \(\ln 10 \int \frac{1}{x \ln x} \, dx\).
Use substitution to solve the integral \(\int \frac{1}{x \ln x} \, dx\). Let \(u = \ln x\), then \(du = \frac{1}{x} dx\), which means \(dx = x \, du\).
Substitute into the integral: \(\int \frac{1}{x u} dx = \int \frac{1}{x u} (x \, du) = \int \frac{1}{u} du\). This integral is \(\ln |u| + C = \ln |\ln x| + C\). Don't forget to multiply back by the constant \(\ln 10\).
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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 of the integrand helps in choosing appropriate substitution or integration techniques.
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Intro to Rational Functions
Logarithmic Functions and Their Properties
Understanding the properties of logarithms, especially change of base formulas and derivatives, is essential. For example, log₁₀(x) can be expressed in terms of natural logarithms, which simplifies integration.
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06:21Properties of Functions
Substitution Method in Integration
This technique involves substituting a part of the integrand with a new variable to simplify the integral. For integrals involving logarithms, substituting the logarithmic expression often reduces the integral to a basic form.
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07:33Euler's Method
Related Practice
Textbook Question
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = 1/x², x > 0
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
Evaluate the integrals in Exercises 33–54.
∫8e^(x+1) dx
Textbook Question
Evaluate the integrals in Exercises 33–54.
∫₀^(π/4) (1 + e^(tan θ)) sec²θ dθ
Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
31. y = cos⁻¹(x) - x sech⁻¹(x)