Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(2θ) cos³(2θ) dθ

검증된 단계별 안내

Start by recognizing the integral: \(\int_0^{\frac{\pi}{2}} \sin^2(2\theta) \cos^3(2\theta) \, d\theta\). Notice that the powers of sine and cosine are both positive integers, which suggests using trigonometric identities or substitution.

Use the substitution \(u = 2\theta\), which implies \(du = 2 \, d\theta\) or \(d\theta = \frac{du}{2}\). Also, change the limits of integration accordingly: when \(\theta = 0\), \(u = 0\); when \(\theta = \frac{\pi}{2}\), \(u = \pi\).

Rewrite the integral in terms of \(u\): \(\int_0^{\pi} \sin^2(u) \cos^3(u) \cdot \frac{1}{2} \, du = \frac{1}{2} \int_0^{\pi} \sin^2(u) \cos^3(u) \, du\).

Express \(\cos^3(u)\) as \(\cos(u) \cdot \cos^2(u)\) and use the Pythagorean identity \(\cos^2(u) = 1 - \sin^2(u)\) to rewrite the integral in terms of \(\sin(u)\): \(\frac{1}{2} \int_0^{\pi} \sin^2(u) \cos(u) (1 - \sin^2(u)) \, du\).

Use the substitution \(t = \sin(u)\), so \(dt = \cos(u) \, du\). The integral becomes \(\frac{1}{2} \int_{t=\sin(0)}^{t=\sin(\pi)} t^2 (1 - t^2) \, dt\). Then, expand the integrand and prepare to integrate term-by-term.

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주요 개념

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

Trigonometric Integrals

Trigonometric integrals involve integrating products or powers of sine and cosine functions. Techniques often include using identities to simplify powers or convert products into sums, making the integral easier to evaluate.

Power-Reducing and Double-Angle Identities

Power-reducing identities express powers of sine and cosine in terms of first powers of cosine with multiple angles. Double-angle identities relate functions like sin(2θ) and cos(2θ) to sin(θ) and cos(θ), helping to simplify integrals involving these expressions.

Substitution Method

The substitution method involves changing variables to simplify an integral. For trigonometric integrals, substituting a function inside sine or cosine (e.g., u = 2θ) can reduce complexity and adjust limits accordingly for definite integrals.

Evaluate the integrals in Exercises 1–22.∫₀^(π/2) sin²(2θ) cos³(2θ) dθ