Textbook Question
Evaluate the integrals in Exercises 111–114.
111. ∫₁^(ln x) (1 / t) dt,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 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 7.7.9
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
9. (sinh(x)+cosh(x))⁴
Verified step by step guidance1
Recall the definitions of hyperbolic sine and cosine in terms of exponentials: \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\) and \(\cosh(x) = \frac{e^{x} + e^{-x}}{2}\).
Add \(\sinh(x)\) and \(\cosh(x)\) using their exponential forms: \(\sinh(x) + \cosh(x) = \frac{e^{x} - e^{-x}}{2} + \frac{e^{x} + e^{-x}}{2}\).
Combine the fractions since they have the same denominator: \(\frac{e^{x} - e^{-x} + e^{x} + e^{-x}}{2} = \frac{2e^{x}}{2} = e^{x}\).
Rewrite the original expression \((\sinh(x) + \cosh(x))^{4}\) as \((e^{x})^{4}\) using the simplification from the previous step.
Apply the exponent rule \((e^{x})^{4} = e^{4x}\) to express the final simplified form in terms of exponentials.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Functions and Their Definitions
Hyperbolic sine (sinh) and cosine (cosh) are defined using exponential functions: sinh(x) = (e^x - e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2. Understanding these definitions allows rewriting expressions involving sinh and cosh in terms of exponentials.
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Definition of the Definite Integral
Algebraic Simplification of Exponential Expressions
After rewriting hyperbolic functions as exponentials, simplifying involves combining like terms, factoring, and applying exponent rules. Mastery of these algebraic techniques is essential to reduce the expression to its simplest form.
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Properties of Exponents and Binomial Expansion
Raising a sum to a power, such as (sinh(x) + cosh(x))^4, often requires using binomial expansion or recognizing patterns in exponentials. Understanding exponent properties helps in expanding and simplifying the expression efficiently.
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Related Practice
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
41. lim (x → 0⁺) (ln x)² / ln(sin x)
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Textbook Question
Verify the integration formulas in Exercises 111–114.
113. ∫ (arcsin x)² dx = x(arcsin x)² - 2x + 2 √(1 - x²) arcsin x + C
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130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?
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90. Find f'(0) for
f(x) = e^(-1/x²), x≠0
= 0, x = 0.
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Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
10. y = ln(t^(3/2))
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