Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (−ln(x)) dx
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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 8.3.58
Evaluate the integrals in Exercises 53–58.
∫ from -π/2 to π/2 of cos(x) cos(7x) dx
Verified step by step guidance1
Recognize that the integral involves the product of two cosine functions: \(\cos(x)\) and \(\cos(7x)\), integrated over the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
Use the product-to-sum identity for cosine functions: \(\cos A \cos B = \frac{1}{2} [\cos(A - B) + \cos(A + B)]\). Applying this, rewrite the integrand as \(\frac{1}{2} [\cos(x - 7x) + \cos(x + 7x)] = \frac{1}{2} [\cos(-6x) + \cos(8x)]\).
Since cosine is an even function, \(\cos(-6x) = \cos(6x)\), so the integrand simplifies to \(\frac{1}{2} [\cos(6x) + \cos(8x)]\).
Split the integral into two separate integrals: \(\frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(6x) \, dx + \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(8x) \, dx\).
Evaluate each integral by finding the antiderivative of cosine, which is sine divided by the coefficient of \(x\), and then apply the limits of integration \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) for each term.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the net area under a curve between two specified limits. It is evaluated by finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower bounds.
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Definition of the Definite Integral
Product-to-Sum Trigonometric Identities
These identities transform products of trigonometric functions into sums or differences, simplifying integration. For example, cos(A)cos(B) = ½[cos(A−B) + cos(A+B)], which helps break down complex integrals into easier terms.
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Symmetry of Trigonometric Functions
Understanding whether a function is even, odd, or neither helps simplify definite integrals over symmetric intervals. Cosine is an even function, so integrals from -a to a can be simplified by doubling the integral from 0 to a.
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Related Practice
Textbook Question
Volume: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.
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Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
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