Textbook Question
Evaluate the integrals in Exercises 31–78.
52. ∫(from 1 to 32)(1/5x) dx
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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 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.PE.55
Evaluate the integrals in Exercises 31–78.
55. ∫(from -2 to -1)e^(-(x+1)) dx
Verified step by step guidance1
Identify the integral to be evaluated: \(\int_{-2}^{-1} e^{-(x+1)} \, dx\).
Rewrite the exponent to clarify the integrand: \(e^{-(x+1)} = e^{-x-1} = e^{-1} \cdot e^{-x}\).
Set up a substitution to simplify the integral. Let \(u = x + 1\), then \(du = dx\). Change the limits accordingly: when \(x = -2\), \(u = -1\); when \(x = -1\), \(u = 0\).
Rewrite the integral in terms of \(u\): \(\int_{u=-1}^{0} e^{-u} \, du\).
Integrate \(e^{-u}\) with respect to \(u\): the antiderivative is \(-e^{-u}\). Then apply the new limits \(u = -1\) to \(u = 0\) to express the definite integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the lower and upper bounds. The result is a number representing the accumulated value of the function over that interval.
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Definition of the Definite Integral
Integration of Exponential Functions
Integrating exponential functions like e^(ax) involves using the rule ∫ e^(ax) dx = (1/a) e^(ax) + C. When the exponent is a linear function of x, substitution may be used to simplify the integral before applying this rule.
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Substitution Method
The substitution method simplifies integrals by changing variables. For an integral involving a composite function, set u equal to the inner function, then rewrite dx in terms of du. This transforms the integral into a simpler form that is easier to evaluate.
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Related Practice
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