Textbook Question
42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.3.28
9–61. Trigonometric integrals Evaluate the following integrals.
28. ∫ 6 sec⁴x dx
Verified step by step guidance1
Step 1: Recognize the integral ∫ 6 sec⁴x dx as a trigonometric integral involving secant raised to the fourth power. The goal is to simplify and evaluate this integral.
Step 2: Recall the standard formula for the integral of sec⁴x: ∫ sec⁴x dx = (1/3) tan³x + tan x + C. This formula can be applied directly to the given integral.
Step 3: Factor out the constant 6 from the integral to simplify the computation: ∫ 6 sec⁴x dx = 6 ∫ sec⁴x dx.
Step 4: Substitute the formula for ∫ sec⁴x dx into the expression: 6 ∫ sec⁴x dx = 6 [(1/3) tan³x + tan x + C].
Step 5: Simplify the expression by distributing the constant 6 across the terms inside the brackets: 6 [(1/3) tan³x + tan x + C] = 2 tan³x + 6 tan x + C.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Trigonometric Functions
Integration of trigonometric functions involves finding the antiderivative of functions that include trigonometric ratios. In this case, the integral of secant raised to a power, such as sec⁴x, requires knowledge of specific integration techniques and formulas related to trigonometric identities.
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Introduction to Trigonometric Functions
Secant Function
The secant function, denoted as sec(x), is the reciprocal of the cosine function, expressed as sec(x) = 1/cos(x). Understanding the properties and behavior of the secant function is crucial for evaluating integrals involving secant, especially when raised to higher powers, as it can often be simplified using trigonometric identities.
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6:22Graphs of Secant and Cosecant Functions
Integration Techniques
Various techniques exist for integrating functions, including substitution, integration by parts, and recognizing patterns in integrals. For sec⁴x, one might use the identity sec²x = 1 + tan²x to rewrite the integral, making it easier to evaluate. Familiarity with these techniques is essential for solving complex integrals.
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06:18Integration by Parts for Definite Integrals
Related Practice
Textbook Question
7–64. Integration review Evaluate the following integrals.
10. ∫ e^(3 - 4x) dx
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Textbook Question
69-72. Volumes of solids Find the volume of the following solids.
70. The region bounded by y = 1/[x²(x² + 2)²], y = 0, x = 1, and x = 2 is revolved about the y-axis.
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Textbook Question
50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:
51. ∫ xⁿ cos(ax) dx = (xⁿ sin(ax))/a - (n/a) ∫ xⁿ⁻¹ sin(ax) dx, for a ≠ 0
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Textbook Question
9–61. Trigonometric integrals Evaluate the following integrals.
37. ∫ [sec⁴(lnθ)]/θ dθ
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Textbook Question
9–61. Trigonometric integrals Evaluate the following integrals.
22. ∫[π/4 to π/2] sin²(2x) cos³(2x) dx
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