92. Evaluate ∫ from 3 to ∞ [ dx / (x √(x² - 9))]

검증된 단계별 안내

Recognize that the integral is an improper integral because the upper limit is infinity. So, rewrite the integral as a limit: \(\displaystyle \lim_{t \to \infty} \int_{3}^{t} \frac{dx}{x \sqrt{x^{2} - 9}}\).

To evaluate the integral \(\int \frac{dx}{x \sqrt{x^{2} - 9}}\), use a trigonometric substitution. Since the integrand contains \(\sqrt{x^{2} - 9}\), let \(x = 3 \sec \theta\), which implies \(dx = 3 \sec \theta \tan \theta \, d\theta\).

Substitute \(x = 3 \sec \theta\) and \(dx\) into the integral. Also, express \(\sqrt{x^{2} - 9}\) in terms of \(\theta\): \(\sqrt{(3 \sec \theta)^{2} - 9} = \sqrt{9 \sec^{2} \theta - 9} = 3 \tan \theta\).

Rewrite the integral in terms of \(\theta\): \(\int \frac{3 \sec \theta \tan \theta \, d\theta}{3 \sec \theta \cdot 3 \tan \theta}\). Simplify the expression by canceling common factors.

After simplification, integrate the resulting expression with respect to \(\theta\). Then, convert back to the variable \(x\) using the inverse trigonometric relationships from the substitution. Finally, apply the limits by converting the original limits \(x=3\) and \(x=t\) to their corresponding \(\theta\) values, and take the limit as \(t \to \infty\).

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주요 개념

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

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, we replace the infinite limit with a variable and take the limit as it approaches infinity, ensuring the integral converges to a finite value.

Trigonometric Substitution

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

Definite Integral Evaluation

Evaluating a definite integral requires finding the antiderivative and then applying the Fundamental Theorem of Calculus by substituting the upper and lower limits. For improper integrals, this includes taking limits at infinite bounds to determine convergence.