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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 8.4.38
Chapter 8, Problem 8.4.38

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (1 - r²)^(5/2) / r⁸ dr

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Identify the integral to solve: \(\int \frac{(1 - r^{2})^{5/2}}{r^{8}} \, dr\).
Recognize that the integrand contains a term of the form \(\sqrt{1 - r^{2}}\) raised to a power, suggesting a trigonometric substitution related to \(r = \sin \theta\) or \(r = \cos \theta\).
Choose the substitution \(r = \sin \theta\), which implies \(dr = \cos \theta \, d\theta\). This substitution transforms \(1 - r^{2}\) into \(1 - \sin^{2} \theta = \cos^{2} \theta\).
Rewrite the integral in terms of \(\theta\) by substituting \(r = \sin \theta\), \(dr = \cos \theta \, d\theta\), and simplifying the powers of trigonometric functions accordingly.
After rewriting, simplify the integral to a form involving powers of sine and cosine functions, then use trigonometric identities or reduction formulas to integrate with respect to \(\theta\).

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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, the integral simplifies into a form involving trigonometric identities, making it easier to integrate.
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Integration of Rational Functions

Integration of rational functions involves integrating expressions where one polynomial is divided by another. Techniques include polynomial division, partial fraction decomposition, or substitutions to simplify the integral into manageable parts.
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Intro to Rational Functions

Power Functions and Exponent Rules

Understanding how to manipulate and integrate power functions, especially with fractional exponents, is essential. Applying exponent rules helps simplify expressions before integration, and recognizing when to use substitution or other methods is key.
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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 0 to ∞ of (dθ / (1 + e^θ))

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Textbook Question

18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:

a. the y-axis.

Textbook Question

Evaluate the integrals in Exercises 33–52.

∫ cot⁶(2x) dx

Textbook Question

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.

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.

∫ (dt / t√(3 + t²)

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Textbook Question

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫ cos t dt / (1 - cos t)