Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 8 cos^4(2πt) dt

검증된 단계별 안내

Recognize that the integral involves a power of cosine: \(\int 8 \cos^4(2\pi t) \, dt\). To use reduction formulas, we want to express \(\cos^4(\theta)\) in terms of lower powers or multiple angles.

Recall the power-reduction formula for cosine: \(\cos^2(x) = \frac{1 + \cos(2x)}{2}\). Since we have \(\cos^4(2\pi t)\), rewrite it as \((\cos^2(2\pi t))^2\).

Apply the power-reduction formula to \(\cos^2(2\pi t)\): \(\cos^2(2\pi t) = \frac{1 + \cos(4\pi t)}{2}\). Then, \(\cos^4(2\pi t) = \left( \frac{1 + \cos(4\pi t)}{2} \right)^2\).

Expand the square: \(\left( \frac{1 + \cos(4\pi t)}{2} \right)^2 = \frac{1}{4} (1 + 2\cos(4\pi t) + \cos^2(4\pi t))\). Now, apply the power-reduction formula again to \(\cos^2(4\pi t)\).

Substitute \(\cos^2(4\pi t) = \frac{1 + \cos(8\pi t)}{2}\) into the expression, simplify the integrand, and then integrate term-by-term with respect to \(t\).

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주요 개념

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Reduction Formulas

Reduction formulas are recursive relationships that express an integral involving a power of a function in terms of an integral with a lower power. They simplify complex integrals by breaking them down step-by-step, often used for powers of trigonometric functions like cosine.

Integration of Powers of Cosine

Integrating powers of cosine functions often requires using trigonometric identities or reduction formulas. For example, cos^n(x) can be expressed in terms of cos^(n-2)(x) and cos(2x), enabling easier integration by reducing the power gradually.

Trigonometric Identities

Trigonometric identities, such as the power-reduction or double-angle formulas, are essential tools for rewriting powers of cosine into integrable forms. For instance, cos^2(x) = (1 + cos(2x))/2 helps transform the integral into simpler terms.

Use reduction formulas to evaluate the integrals in Exercises 41–50.∫ 8 cos^4(2πt) dt