Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
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.RE.2
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.2Chapter 7, Problem 7.RE.2
2–9. Integrals Evaluate the following integrals.
∫ (eˣ / (4eˣ + 6)) dx
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{e^{x}}{4e^{x} + 6} \, dx\).
Notice that the denominator is a linear function of \(e^{x}\). This suggests using a substitution where \(u = 4e^{x} + 6\).
Compute the derivative of \(u\) with respect to \(x\): \(\frac{du}{dx} = 4e^{x}\). Rearranging, we get \(e^{x} dx = \frac{du}{4}\).
Rewrite the integral in terms of \(u\): replace \(e^{x} dx\) with \(\frac{du}{4}\) and the denominator with \(u\), so the integral becomes \(\int \frac{1}{u} \cdot \frac{du}{4} = \frac{1}{4} \int \frac{1}{u} \, du\).
Integrate \(\frac{1}{u}\) with respect to \(u\) to get \(\ln|u|\), then substitute back \(u = 4e^{x} + 6\) to express the answer in terms of \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand whose derivative also appears in the integral, allowing the integral to be rewritten in terms of a new variable, making it easier to solve.
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Substitution With an Extra Variable
Exponential Functions
Exponential functions have the form e^x, where e is Euler's number. Their derivatives and integrals are unique because the derivative of e^x is itself, which often simplifies integration problems involving exponential terms.
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Rational Functions in Integration
Rational functions are ratios of polynomials or expressions involving variables. Integrating rational functions often requires algebraic manipulation or substitution to rewrite the integral into a more manageable form, especially when the denominator contains expressions related to the numerator.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
A power line is attached at the same height to two utility poles that are separated by a distance of 100 ft; the power line follows the curve ƒ(x) = a cosh x/a. Use the following steps to find the value of a that produces a sag of 10 ft midway between the poles. Use a coordinate system that places the poles at x = ±50.
a. Show that a satisfies the equation cosh 50/a − 1 = 10/a.
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Textbook Question
2–9. Integrals Evaluate the following integrals.
∫ (x + 4) / (x² + 8x + 25) dx
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Textbook Question
2–9. Integrals Evaluate the following integrals.
∫₀¹ (x² / (9 − x⁶)) dx
Textbook Question
Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
a. cosh 0
Textbook Question
Falling body When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after t seconds is given by d(t) = (m/k) ln (cosh (√(kg/m) t)), where m is the mass of the object in kilograms, g = 9.8 m/s² is the acceleration due to gravity, and k is a physical constant.
a. A BASE jumper (m = 75 kg) leaps from a tall cliff and performs a ten-second delay (she free-falls for 10 s and then opens her chute). How far does she fall in 10 s? Assume k = 0.2.