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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.1.65d
Chapter 8, Problem 8.1.65d

 Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. ∫(1/eˣ) dx = ln eˣ + C.

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1
Step 1: Analyze the given integral ∫(1/eˣ) dx. The integrand is 1/eˣ, which can be rewritten as e⁻ˣ using the property of exponents.
Step 2: Recall the standard rule for integrating exponential functions. The integral of e⁻ˣ with respect to x is ∫e⁻ˣ dx = -e⁻ˣ + C, where C is the constant of integration.
Step 3: Compare the given statement ∫(1/eˣ) dx = ln eˣ + C with the correct integral result. Note that ln eˣ simplifies to x because the natural logarithm and exponential functions are inverses of each other.
Step 4: Observe that the given statement suggests the integral of 1/eˣ is ln eˣ + C, which simplifies to x + C. This is incorrect because the integral of 1/eˣ is actually -e⁻ˣ + C, not x + C.
Step 5: Conclude that the statement ∫(1/eˣ) dx = ln eˣ + C is false. Provide the correct result, ∫(1/eˣ) dx = -e⁻ˣ + C, as a counterexample to demonstrate why the given statement is incorrect.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of determining the area under a curve represented by a function over a specified interval. Understanding integration is crucial for evaluating definite and indefinite integrals, which are essential for solving problems related to areas, volumes, and other applications in mathematics and physics.
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Integration by Parts for Definite Integrals

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is an important function in calculus, particularly in integration and differentiation. The natural logarithm has unique properties, such as ln(e^x) = x, which are useful for simplifying expressions and solving equations involving exponential functions.
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Derivative of the Natural Logarithmic Function

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where a and b are constants, and e is the base of the natural logarithm. These functions are characterized by their rapid growth or decay and are widely used in various fields, including finance, biology, and physics. Understanding the properties of exponential functions is essential for evaluating integrals involving e^x and applying the rules of integration correctly.
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Related Practice
Textbook Question

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.

The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]

(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project "The exponential Eiffel Tower")

d. Prove this property holds for any a ≥ 0 and c > 0:

The tangent lines to y = ±e^(-c·x) at x = a always intersect at R's center of mass

(Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique, 332, 571-584, 2004)

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Textbook Question

109. Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by W = ∫ from R to ∞ of F(x) dx, where R = 6370 km is the approximate radius of Earth, F(x) = (GMm)/x² is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 × 10¹⁴ m³/s².

c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R ≤ 2GM/c². For Earth to be a black hole, what would its radius need to be?

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

69. Let f(x) = sin(eˣ).

d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1.

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Textbook Question

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

47. ∫(1 to e) (1/x) dx; n = 50

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

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Textbook Question

101. Many methods needed Show that the integral from ∫(from 0 to ∞)(sqrt(x) * ln x) / (1 + x)^2 dx equals pi, following these steps

d. Evaluate the remaining integral using the change of variables z = sqrt(x)

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Textbook Question

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.

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