Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(x) = ln(3 sin² 4x)
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.41
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.41Chapter 7, Problem 7.3.41
37–56. Integrals Evaluate each integral.
∫ tanh²x dx (Hint: Use an identity.)
Verified step by step guidance1
Recall the identity for hyperbolic tangent squared: \(\tanh^{2}x = 1 - \sech^{2}x\). This will help simplify the integral.
Rewrite the integral using the identity: \(\int \tanh^{2}x \, dx = \int (1 - \sech^{2}x) \, dx\).
Split the integral into two separate integrals: \(\int 1 \, dx - \int \sech^{2}x \, dx\).
Integrate each term separately: The integral of 1 with respect to \(x\) is \(x\), and the integral of \(\sech^{2}x\) is \(\tanh x\).
Combine the results to write the integral as \(x - \tanh x + C\), where \(C\) is the constant of integration.
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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 Identities
Hyperbolic functions like tanh(x) are analogs of trigonometric functions but based on exponential functions. Key identities, such as tanh²x + sech²x = 1, help simplify expressions and integrals involving these functions.
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Verifying Trig Equations as Identities
Integration Techniques Using Identities
Using algebraic or trigonometric/hyperbolic identities can transform complicated integrals into simpler forms. For example, rewriting tanh²x using an identity allows the integral to be expressed in terms of easier-to-integrate functions.
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Basic Integration of Hyperbolic Functions
Integrals involving hyperbolic functions often reduce to standard forms, such as ∫sech²x dx = tanh x + C. Recognizing these standard integrals is essential for solving problems efficiently.
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Related Practice
Textbook Question
63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₁/₈¹ dx/x√(1 + x²/³)
Textbook Question
37–56. Integrals Evaluate each integral.
∫ dx/x√(16 + x²)
Textbook Question
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ (x²) / (4x³ + 7) dx
Textbook Question
Tsunamis A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging from 150 to 1000 km. Imagine a tsunami traveling across the Pacific Ocean, which is the deepest ocean in the world, with an average depth of about 4000 m. Explain why the shallow-water velocity equation (Exercise 75) applies to tsunamis even though the actual depth of the water is large. What does the shallow-water equation say about the speed of a tsunami in the Pacific Ocean (use d = 4000 m)?
Textbook Question
37–56. Integrals Evaluate each integral.
∫ sech² w tanh w dw
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