Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
39. lim (x → ∞) (ln 2x - ln(x + 1))
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 7.2.52
Evaluate the integrals in Exercises 39–56.
52. ∫(from π/4 to π/2)cot(t)dt
Verified step by step guidance1
Recall the definition of the cotangent function: \(\cot(t) = \frac{\cos(t)}{\sin(t)}\).
Recognize that the integral of \(\cot(t)\) can be expressed using a standard antiderivative: \(\int \cot(t) \, dt = \ln|\sin(t)| + C\).
Set up the definite integral using the antiderivative: \(\int_{\pi/4}^{\pi/2} \cot(t) \, dt = \left[ \ln|\sin(t)| \right]_{\pi/4}^{\pi/2}\).
Evaluate the antiderivative at the upper limit \(t = \pi/2\) and the lower limit \(t = \pi/4\): calculate \(\ln|\sin(\pi/2)|\) and \(\ln|\sin(\pi/4)|\) separately.
Subtract the value of the antiderivative at the lower limit from the value at the upper limit to find the value of the definite integral: \(\ln|\sin(\pi/2)| - \ln|\sin(\pi/4)|\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition and Properties of the Cotangent Function
Cotangent, cot(t), is the ratio of cosine to sine, cot(t) = cos(t)/sin(t). Understanding its behavior and domain is essential, especially since cotangent has discontinuities where sin(t) = 0. This helps in evaluating integrals involving cotangent and recognizing potential issues with limits.
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Properties of Functions
Integration of Trigonometric Functions
Integrating cotangent involves rewriting it in terms of sine and cosine or using known integral formulas. The integral of cot(t) is ln|sin(t)| + C, which is crucial for solving definite integrals involving cotangent over a given interval.
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Evaluating Definite Integrals with Logarithmic Functions
After integrating, evaluating the definite integral requires substituting the limits into the antiderivative, often involving logarithmic expressions. Careful handling of absolute values and domain restrictions ensures correct evaluation of the integral's value.
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Related Practice
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Evaluate the integrals in Exercises 39–56.
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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.
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Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
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