Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / (x³ √(x² - 1)), where x > 1

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Identify the integral to solve: \(\int \frac{2 \, dx}{x^{3} \sqrt{x^{2} - 1}}\) with the condition \(x > 1\).

Recognize that the integrand contains \(\sqrt{x^{2} - 1}\), which suggests using a trigonometric substitution such as \(x = \sec(\theta)\) because \(\sec^{2}(\theta) - 1 = \tan^{2}(\theta)\).

Perform the substitution: let \(x = \sec(\theta)\), then compute \(dx = \sec(\theta) \tan(\theta) \, d\theta\). Also, rewrite the expressions inside the integral in terms of \(\theta\).

Rewrite the integral entirely in terms of \(\theta\) by substituting \(x\), \(dx\), and \(\sqrt{x^{2} - 1}\), then simplify the resulting expression to a form that is easier to integrate.

Integrate with respect to \(\theta\), then back-substitute using \(\theta = \sec^{-1}(x)\) to express the answer in terms of \(x\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Integration Techniques for Rational Functions

This involves methods to integrate functions expressed as ratios of polynomials or involving roots. Recognizing the form helps decide whether substitution, partial fractions, or trigonometric substitution is appropriate to simplify the integral.

Trigonometric Substitution

Trigonometric substitution is used to simplify integrals containing expressions like √(x² - a²). By substituting x = a sec(θ), the radical simplifies using trigonometric identities, making the integral easier to evaluate.

Domain Considerations and Restrictions

Understanding the domain (x > 1) is crucial because it affects the choice of substitution and the sign of expressions like √(x² - 1). It ensures the substitution is valid and the integral is evaluated correctly within the given constraints.

Evaluate the integrals in Exercises 1–14.∫ (2 dx) / (x³ √(x² - 1)), where x > 1