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

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Recall the reduction formula for powers of cotangent: for an integer \(n \geq 2\), the integral \(\int \cot^n(t) \, dt\) can be expressed in terms of \(\int \cot^{n-2}(t) \, dt\). Specifically, use the identity \(\cot^2(t) = \csc^2(t) - 1\) to rewrite higher powers.

Rewrite \(\cot^4(t)\) as \((\cot^2(t))^2\) and then express \(\cot^2(t)\) in terms of \(\csc^2(t)\): \(\cot^4(t) = (\csc^2(t) - 1)^2\).

Expand the square to get \(\cot^4(t) = \csc^4(t) - 2\csc^2(t) + 1\), so the integral becomes \(\int 8 \cot^4(t) \, dt = 8 \int (\csc^4(t) - 2\csc^2(t) + 1) \, dt\).

Split the integral into three separate integrals: \(8 \int \csc^4(t) \, dt - 16 \int \csc^2(t) \, dt + 8 \int 1 \, dt\).

Use known reduction formulas or standard integrals for \(\int \csc^2(t) \, dt\) and \(\int \csc^4(t) \, dt\) to evaluate each term, then combine the results to express the original integral.

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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, making it easier to evaluate powers of trigonometric functions like cotangent.

Integration of Trigonometric Functions

Integrating powers of trigonometric functions such as cotangent requires understanding their identities and derivatives. Recognizing how to rewrite cot^n(t) using identities or expressing it in terms of sine and cosine helps in applying integration techniques effectively.

Trigonometric Identities

Trigonometric identities, like cot^2(t) = csc^2(t) - 1, are essential tools for simplifying integrals involving powers of cotangent. These identities allow the integral to be rewritten in a more manageable form, facilitating the use of reduction formulas or direct integration.

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