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

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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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주요 개념

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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.

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.

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.

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