Evaluate the integrals in Exercises 23–32.
∫₋π^π (1 - cos²(t))^(3/2) dt

검증된 단계별 안내

Recognize that the integrand is \( (1 - \cos^{2}(t))^{3/2} \). Recall the Pythagorean identity \( \sin^{2}(t) + \cos^{2}(t) = 1 \), which allows us to rewrite the integrand in terms of \( \sin(t) \).

Rewrite the integrand as \( (\sin^{2}(t))^{3/2} = |\sin(t)|^{3} \) because \( (\sin^{2}(t))^{3/2} = (\sin^{2}(t))^{1 \cdot 3/2} = |\sin(t)|^{3} \).

Since the integral is from \( -\pi \) to \( \pi \), and \( |\sin(t)|^{3} \) is an even function (because \( |\sin(-t)| = |\sin(t)| \)), use the property of even functions to simplify the integral: \( \int_{-\pi}^{\pi} |\sin(t)|^{3} dt = 2 \int_{0}^{\pi} |\sin(t)|^{3} dt \).

On the interval \( [0, \pi] \), \( \sin(t) \) is non-negative, so \( |\sin(t)|^{3} = \sin^{3}(t) \). Thus, the integral becomes \( 2 \int_{0}^{\pi} \sin^{3}(t) dt \).

To evaluate \( \int \sin^{3}(t) dt \), use the reduction formula or rewrite \( \sin^{3}(t) \) as \( \sin(t) \cdot \sin^{2}(t) = \sin(t)(1 - \cos^{2}(t)) \), then use substitution \( u = \cos(t) \) to integrate.

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주요 개념

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing that 1 - cos²(t) equals sin²(t) simplifies the integrand, making the integral easier to evaluate.

Definite Integrals and Symmetry

Definite integrals calculate the net area under a curve between two limits. When the integrand is an even or odd function, symmetry properties can simplify the integral, such as doubling the integral from 0 to π if the function is even.

Power Functions and Integration Techniques

Integrating functions raised to a power often requires substitution or recognizing standard integral forms. Here, raising sin(t) to the third power involves understanding how to integrate powers of sine, possibly using reduction formulas or substitution.

Evaluate the integrals in Exercises 23–32.∫₋π^π (1 - cos²(t))^(3/2) dt