Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ sec³(x) tan³(x) dx
Ch. 8 - Techniques of Integration
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 8, Problem 8.8.26
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (−ln(x)) dx
Verified step by step guidance1
Recognize that the integral is \( \int_0^1 (-\ln(x)) \, dx \). Since the integrand is \( -\ln(x) \), rewrite the integral as \( - \int_0^1 \ln(x) \, dx \) to simplify the process.
Recall that the integral of \( \ln(x) \) can be found using integration by parts. Set \( u = \ln(x) \) and \( dv = dx \). Then, compute \( du = \frac{1}{x} dx \) and \( v = x \).
Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the values to get \( \int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx = x \ln(x) - \int 1 \, dx \).
Simplify the integral to \( x \ln(x) - x + C \). Now, evaluate the definite integral \( \int_0^1 \ln(x) \, dx = [x \ln(x) - x]_0^1 \).
Evaluate the limits carefully: at \( x=1 \), \( 1 \cdot \ln(1) - 1 = 0 - 1 = -1 \); at \( x=0 \), use the limit \( \lim_{x \to 0^+} x \ln(x) = 0 \), so the expression at 0 is \( 0 - 0 = 0 \). Therefore, \( \int_0^1 \ln(x) \, dx = -1 \). Finally, multiply by \( -1 \) to get the value of the original integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals and Convergence
An integral is improper if the interval is infinite or the integrand is unbounded. Here, the integrand involves ln(x), which is unbounded near 0, so we consider the limit as x approaches 0 to check convergence before evaluating.
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Improper Integrals: Infinite Intervals
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, often used when integrating functions like ln(x) or products involving logarithms.
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Properties of the Natural Logarithm Function
The natural logarithm ln(x) is defined for x > 0 and approaches negative infinity as x approaches 0 from the right. Understanding its behavior near zero is crucial for evaluating integrals involving ln(x), especially when combined with limits.
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Related Practice
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