Textbook Question
Evaluate the integrals in Exercises 77–90.
81. ∫dy/(y²-2y+5)
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.2.73
73. Find the area between the curves y=ln(x) and y=ln(2x) from x=1 to x=5.
Verified step by step guidance1
Identify the two functions given: \( y = \ln(x) \) and \( y = \ln(2x) \). We want to find the area between these curves from \( x = 1 \) to \( x = 5 \).
Determine which function is on top and which is on the bottom in the interval \( [1, 5] \). Since \( \ln(2x) = \ln(2) + \ln(x) \), it is always greater than \( \ln(x) \) for \( x > 0 \). So, \( y = \ln(2x) \) is the upper curve and \( y = \ln(x) \) is the lower curve.
Set up the integral for the area between the curves as \( \int_1^5 [\ln(2x) - \ln(x)] \, dx \).
Simplify the integrand using logarithm properties: \( \ln(2x) - \ln(x) = \ln\left(\frac{2x}{x}\right) = \ln(2) \). So the integral becomes \( \int_1^5 \ln(2) \, dx \).
Evaluate the integral by integrating the constant \( \ln(2) \) over \( [1, 5] \), which is \( \ln(2) \times (5 - 1) \). This gives the area between the curves.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals and Area Between Curves
The area between two curves over an interval is found by integrating the difference of their functions. Specifically, the integral of the upper function minus the lower function from the lower to upper bounds gives the enclosed area.
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Finding Area Between Curves on a Given Interval
Properties of Logarithmic Functions
Understanding logarithmic functions, such as ln(x) and ln(2x), is essential. Using log properties like ln(2x) = ln(2) + ln(x) helps simplify the integrand and makes the integration process more straightforward.
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Evaluating Definite Integrals Involving Logarithms
Integrating logarithmic functions often requires integration by parts or recognizing standard integral forms. Evaluating the definite integral involves substituting the limits after integration to find the exact area.
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Related Practice
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49. ∫3sec²t/(6 + 3tan(t)) dt
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Textbook Question
In Exercises 73 and 74, repeat the steps above to solve for the functions y=f(x) and x=f^(-1)(y) defined implicitly by the given equations over the interval.
73. y^(1/3) - 1 = (x+2)³, -5 ≤ x ≤ 5, x_0 = -3/2