Textbook Question
82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.
84. ∫ (from 0 to π) sec²x dx*(Note: Potential improperness at x = π/2)*
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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.R.79
76-81. Table of integrals Use a table of integrals to evaluate the following integrals.
79. ∫ sec⁵x dx
Verified step by step guidance1
Step 1: Recognize that the integral ∫ sec⁵x dx matches a form found in a table of integrals. Tables of integrals often provide solutions for powers of secant functions.
Step 2: Locate the formula in the table of integrals for ∫ sec⁵x dx. Typically, this formula involves breaking down the integral into simpler terms using reduction formulas or trigonometric identities.
Step 3: Apply the formula from the table of integrals. For example, the formula might express ∫ sec⁵x dx in terms of ∫ sec³x dx and ∫ secx dx, which are simpler integrals.
Step 4: Substitute the expressions for ∫ sec³x dx and ∫ secx dx into the formula, if required, and simplify the result. This step may involve algebraic manipulation or substitution.
Step 5: Add the constant of integration (C) to the final expression, as indefinite integrals always include an arbitrary constant.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration
Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. Understanding the basic rules and techniques of integration is essential for solving integral problems.
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Integration by Parts for Definite Integrals
Secant Function
The secant function, denoted as sec(x), is defined as the reciprocal of the cosine function: sec(x) = 1/cos(x). It is important in calculus, particularly in integration and differentiation, as it appears frequently in trigonometric integrals. Recognizing the properties and behavior of the secant function helps in evaluating integrals involving secant and its powers.
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6:22Graphs of Secant and Cosecant Functions
Table of Integrals
A table of integrals is a reference tool that lists common integrals and their corresponding results. It simplifies the process of integration by providing ready-made solutions for frequently encountered functions, such as polynomials, trigonometric functions, and exponential functions. Familiarity with a table of integrals can significantly speed up the evaluation of complex integrals, like ∫ sec⁵x dx.
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Related Practice
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
38. ∫ (from π/4 to π/2) x csc²x dx
Textbook Question
Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
b. To evaluate the integral ∫dx/√(x² − 100) analytically, it is best to use partial fractions.
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Textbook Question
Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
d. ∫2 sin x cos x dx = −(1/2) cos 2x + C.
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Textbook Question
95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.
96. ∫ (from 1 to 3) dx/(x³ + x + 1); n = 4
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
9. ∫ (from 0 to π/4) cos⁵ 2x sin² 2x dx