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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.4.24
Chapter 8, Problem 8.4.24

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ √(9 - w²) dw / w²

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Identify the integral to solve: \(\int \frac{\sqrt{9 - w^{2}}}{w^{2}} \, dw\).
Recognize that the integrand contains a square root of the form \(\sqrt{a^{2} - w^{2}}\), which suggests using a trigonometric substitution: let \(w = 3 \sin(\theta)\), where \(a = 3\).
Compute the differential: \(dw = 3 \cos(\theta) \, d\theta\). Also, rewrite the expressions inside the integral in terms of \(\theta\): \(\sqrt{9 - w^{2}} = \sqrt{9 - 9 \sin^{2}(\theta)} = 3 \cos(\theta)\) and \(w^{2} = 9 \sin^{2}(\theta)\).
Substitute all parts into the integral to express it entirely in terms of \(\theta\): replace \(\sqrt{9 - w^{2}}\) with \(3 \cos(\theta)\), \(w^{2}\) with \(9 \sin^{2}(\theta)\), and \(dw\) with \(3 \cos(\theta) \, d\theta\). This will simplify the integral to a trigonometric integral.
Simplify the resulting integral and use trigonometric identities to integrate with respect to \(\theta\). After integrating, substitute back \(\theta = \arcsin(\frac{w}{3})\) to express the answer in terms of \(w\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Substitution

Trigonometric substitution is a technique used to evaluate integrals involving expressions like √(a² - x²), √(a² + x²), or √(x² - a²). By substituting x with a trigonometric function (e.g., x = a sin θ), the integral simplifies using trigonometric identities, making it easier to solve.
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Integration of Rational Functions

Integration of rational functions involves integrating expressions where the integrand is a ratio of polynomials or functions. Recognizing when to simplify or rewrite the integrand, such as separating terms or using substitution, is essential for solving integrals involving algebraic expressions.
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Use of Pythagorean Identities

Pythagorean identities like sin²θ + cos²θ = 1 are fundamental in trigonometric substitution. They allow the conversion of square root expressions into simpler trigonometric forms, facilitating the integration process by reducing complex radicals to manageable trigonometric expressions.
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Related Practice
Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from e to e^e of (ln(ln x) dx)

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Textbook Question

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.

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Textbook Question

Evaluate the integrals in Exercises 1–24 using integration by parts.

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Textbook Question

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ csc³(√θ) / √θ dθ

Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

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Textbook Question

Evaluate the integrals in Exercises 23–32.

∫₀^(π/6) √(1 + sin(x)) dx

(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))