Evaluate the integrals in Exercises 1–14.
∫ √(1 - 9t²) dt

검증된 단계별 안내

Recognize that the integral \( \int \sqrt{1 - 9t^2} \, dt \) resembles the form \( \int \sqrt{a^2 - u^2} \, du \), which suggests using a trigonometric substitution to simplify the square root expression.

Set the substitution \( u = 3t \), so that \( u^2 = 9t^2 \). Then, rewrite the integral in terms of \( u \): \( \int \sqrt{1 - u^2} \cdot \frac{du}{3} \). This simplifies the integral to \( \frac{1}{3} \int \sqrt{1 - u^2} \, du \).

Use the trigonometric substitution \( u = \sin \theta \), which implies \( du = \cos \theta \, d\theta \). This transforms the integral into \( \frac{1}{3} \int \sqrt{1 - \sin^2 \theta} \cdot \cos \theta \, d\theta \).

Simplify the square root using the Pythagorean identity: \( \sqrt{1 - \sin^2 \theta} = \cos \theta \). The integral becomes \( \frac{1}{3} \int \cos \theta \cdot \cos \theta \, d\theta = \frac{1}{3} \int \cos^2 \theta \, d\theta \).

Use the power-reduction formula for \( \cos^2 \theta \): \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \). Substitute this into the integral and integrate with respect to \( \theta \). After integrating, substitute back \( \theta = \arcsin u \) and then \( u = 3t \) to express the answer in terms of \( t \).

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주요 개념

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

Trigonometric Substitution

Trigonometric substitution is a technique used to evaluate integrals involving square roots of quadratic expressions. By substituting a trigonometric function for the variable, the integral simplifies using trigonometric identities. For example, for integrals with √(a² - x²), substituting x = a sin(θ) helps transform the integral into a trigonometric form.

Integration of Trigonometric Functions

After substitution, the integral often involves trigonometric functions like sine and cosine. Understanding how to integrate these functions, including using identities and standard integral formulas, is essential. This allows the integral to be evaluated in terms of θ before converting back to the original variable.

Back-Substitution

Once the integral is evaluated in terms of the trigonometric variable, back-substitution replaces the trigonometric expression with the original variable. This step uses the initial substitution relationship and trigonometric identities to express the final answer in terms of the original variable t.

Evaluate the integrals in Exercises 1–14.∫ √(1 - 9t²) dt