Textbook Question
Center of gravity: Find the center of gravity of the region bounded by the x-axis, the curve y = sec x, and the lines x = -pi/4 and x = pi/4.
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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 8.8.18
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / [x√(x² − 1)]
Verified step by step guidance1
Identify the integral to be evaluated: \(\displaystyle \int_1^{\infty} \frac{dx}{x \sqrt{x^2 - 1}}\).
Recognize that the integrand involves \(\sqrt{x^2 - 1}\), which suggests a trigonometric substitution such as \(x = \sec(\theta)\), because \(\sec^2(\theta) - 1 = \tan^2(\theta)\).
Perform the substitution \(x = \sec(\theta)\), then compute \(dx = \sec(\theta) \tan(\theta) d\theta\). Also, rewrite the integrand in terms of \(\theta\):
\[\frac{1}{x \sqrt{x^2 - 1}} dx = \frac{1}{\sec(\theta) \sqrt{\sec^2(\theta) - 1}} \cdot \sec(\theta) \tan(\theta) d\theta.\]
Simplify the expression inside the integral using the identity \(\sqrt{\sec^2(\theta) - 1} = \tan(\theta)\), and then simplify the integrand to a function of \(\theta\) that is easier to integrate.
Change the limits of integration from \(x\) to \(\theta\) using \(x = \sec(\theta)\), then integrate with respect to \(\theta\). Finally, substitute back to \(x\) to express the answer in terms of the original variable.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals
Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, we replace the infinite limit with a variable, compute the integral, and then take the limit as the variable approaches infinity to determine convergence and value.
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11:11
Improper Integrals: Infinite Intervals
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integrand into a more manageable form. Choosing an appropriate substitution, such as a trigonometric or hyperbolic function, can help evaluate integrals involving expressions like √(x² − 1).
Recommended video:
07:33Euler's Method
Trigonometric Identities and Inverse Functions
Trigonometric identities relate expressions involving squares and roots, such as x² − 1, to trigonometric functions like secant and tangent. Recognizing these allows rewriting the integral in terms of inverse trigonometric functions, facilitating evaluation without tables.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 3 sec^4(3x) dx
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ x² sin(x) dx
Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)
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Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2 ln(z³)) / (16z) dz
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Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ + 4)) dx