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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 8.4.32
Chapter 8, Problem 8.4.32

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 - x²)^(1/2) / x⁴ dx

Verified step by step guidance
1
Identify the integral to solve: \(\int \frac{\sqrt{1 - x^{2}}}{x^{4}} \, dx\).
Recognize that the integrand contains \(\sqrt{1 - x^{2}}\), which suggests a trigonometric substitution using \(x = \sin \theta\) because \(1 - \sin^{2} \theta = \cos^{2} \theta\).
Make the substitution \(x = \sin \theta\), then compute \(dx = \cos \theta \, d\theta\). Rewrite the integral in terms of \(\theta\):
\[\int \frac{\sqrt{1 - \sin^{2} \theta}}{\sin^{4} \theta} \cdot \cos \theta \, d\theta = \int \frac{\cos \theta}{\sin^{4} \theta} \cdot \cos \theta \, d\theta = \int \frac{\cos^{2} \theta}{\sin^{4} \theta} \, d\theta.\]
Simplify the integral to \(\int \frac{\cos^{2} \theta}{\sin^{4} \theta} \, d\theta\). Use the identity \(\cos^{2} \theta = 1 - \sin^{2} \theta\) to rewrite the numerator if needed, and express the integral in terms of powers of \(\sin \theta\) to facilitate integration.
After integrating with respect to \(\theta\), substitute back \(\theta = \arcsin x\) to express the answer in terms of \(x\).

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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), √(x² + a²), or √(x² - a²). By substituting x with a trigonometric function (e.g., x = a sin θ), the integral is transformed into a trigonometric integral that is often easier to solve.
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Introduction to Trigonometric Functions

Integration of Rational Functions

Integration of rational functions involves integrating expressions where the integrand is a ratio of polynomials. Recognizing when to simplify or rewrite the integrand, such as expressing powers of x in the denominator, helps in applying substitution or partial fractions to evaluate the integral.
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Intro to Rational Functions

Simplifying Radicals in Integrals

Simplifying radicals like √(1 - x²) is essential before integration. This often involves rewriting the expression using trigonometric identities or algebraic manipulation to make the integral more manageable, especially when combined with powers of x in the denominator.
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Limits of Rational Functions with Radicals
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