Textbook Question
19. Show that e^x grows faster as x→∞ than x^n for any positive integer n, even x^1,000,000. (Hint: What is the nth derivative of x^n?)
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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 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 7.2.39
Evaluate the integrals in Exercises 39–56.
39. ∫(from -3 to -2)dx/x
Verified step by step guidance1
Identify the integral to be evaluated: \(\int_{-3}^{-2} \frac{1}{x} \, dx\).
Recognize that the integrand \(\frac{1}{x}\) is a standard function whose antiderivative is \(\ln|x|\).
Write the antiderivative: \(F(x) = \ln|x| + C\).
Apply the Fundamental Theorem of Calculus by evaluating \(F(x)\) at the upper and lower limits: compute \(F(-2) - F(-3)\).
Express the definite integral as \(\ln|-2| - \ln|-3|\) and simplify using logarithm properties if needed.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals
An integral is improper if the integrand is undefined or unbounded within the interval of integration. Since the function 1/x is undefined at x = 0, and the interval [-3, -2] does not include zero, the integral is proper here. However, understanding improper integrals is essential when the interval includes points where the function is undefined.
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Improper Integrals: Infinite Intervals
Definite Integral of 1/x
The integral of 1/x with respect to x is ln|x| + C. When evaluating a definite integral of 1/x over an interval that does not cross zero, you compute the difference of the natural logarithms of the absolute values of the endpoints. This requires careful attention to the domain and sign of x.
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Properties of the Natural Logarithm Function
The natural logarithm function ln(x) is defined for positive x and has properties such as ln(a) - ln(b) = ln(a/b). When integrating 1/x, the absolute value ensures the argument of ln is positive, allowing evaluation over negative intervals by considering ln|x|. Understanding these properties helps correctly evaluate integrals involving 1/x.
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Related Practice
Textbook Question
47. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery?
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Textbook Question
Solve the differential equation in Exercises 9–22.
9. 2√(xy) (dy/dx) = 1, x, y > 0
Textbook Question
Evaluate the integrals in Exercises 53–76.
67. ∫dx/(2+(x-1)²)
Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
(-3/2)csc²x(3x/2)
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