Textbook Question
Evaluate the integrals in Exercises 39–56.
55. ∫dx/(2√x + 2x)
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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 7.7.57
Evaluate the integrals in Exercises 41–60.
57. ∫(from 1 to 2)cosh(ln t)/t dt
Verified step by step guidance1
Recognize the integral to evaluate: \(\int_{1}^{2} \frac{\cosh(\ln t)}{t} \, dt\).
Recall the definition of hyperbolic cosine: \(\cosh x = \frac{e^{x} + e^{-x}}{2}\). Substitute \(x = \ln t\) to rewrite the integrand.
Rewrite the integrand as \(\frac{1}{t} \cdot \frac{e^{\ln t} + e^{-\ln t}}{2} = \frac{1}{2t} (t + \frac{1}{t})\) because \(e^{\ln t} = t\) and \(e^{-\ln t} = \frac{1}{t}\).
Simplify the expression inside the integral to \(\frac{1}{2t} (t + \frac{1}{t}) = \frac{1}{2} \left(1 + \frac{1}{t^{2}}\right)\).
Split the integral into two simpler integrals: \(\frac{1}{2} \int_{1}^{2} 1 \, dt + \frac{1}{2} \int_{1}^{2} \frac{1}{t^{2}} \, dt\), then proceed to integrate each term separately.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Cosine Function (cosh)
The hyperbolic cosine function, cosh(x), is defined as (e^x + e^(-x))/2. It is similar to the cosine function but based on exponential functions. Understanding its properties helps simplify expressions involving cosh, especially when combined with logarithmic arguments.
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Graph of Sine and Cosine Function
Logarithmic Functions and Their Properties
The natural logarithm function, ln(t), is the inverse of the exponential function e^x. Key properties, such as ln(ab) = ln(a) + ln(b) and ln(t^n) = n ln(t), allow simplification of integrands involving ln(t). Recognizing how ln(t) interacts with other functions is crucial for integration.
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Integration Techniques and Substitution
Integration by substitution is a method to simplify integrals by changing variables. Here, substituting u = ln(t) transforms the integral into a more manageable form. Mastery of substitution and recognizing when to apply it is essential for evaluating integrals involving composite functions.
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Related Practice
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