Evaluate the integrals in Exercises 33–52.
∫ 8 cot⁴(t) dt

검증된 단계별 안내

Recall that \( \cot(t) = \frac{\cos(t)}{\sin(t)} \) and that powers of cotangent can be expressed in terms of cosecant using the identity \( \cot^2(t) = \csc^2(t) - 1 \). This will help simplify the integral.

Rewrite \( \cot^4(t) \) as \( (\cot^2(t))^2 \), then use the identity to express it in terms of \( \csc^2(t) \): \[ 8 \cot^4(t) = 8 (\cot^2(t))^2 = 8 (\csc^2(t) - 1)^2 \]

Expand the square to get: \[ 8 (\csc^4(t) - 2 \csc^2(t) + 1) \] which breaks the integral into three simpler integrals:

Set up the integral as the sum of three integrals: \[ \int 8 \cot^4(t) \, dt = \int 8 \csc^4(t) \, dt - \int 16 \csc^2(t) \, dt + \int 8 \, dt \]

Evaluate each integral separately: - For \( \int \csc^2(t) \, dt \), recall that the derivative of \( -\cot(t) \) is \( \csc^2(t) \). - For \( \int \csc^4(t) \, dt \), use reduction formulas or rewrite \( \csc^4(t) = (\csc^2(t))^2 \) and express in terms of \( \cot(t) \) and \( \csc(t) \) to integrate. - The integral of a constant is straightforward. Combine all results to express the final integral.

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주요 개념

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. For integrating powers of cotangent, identities like cot²(t) = csc²(t) - 1 help simplify the integrand into more manageable terms.

Integration of Powers of Trigonometric Functions

Integrating powers of trigonometric functions often requires rewriting the integrand using identities or reduction formulas. For cotangent raised to a power, expressing cot⁴(t) in terms of cot²(t) and then using substitution or known integrals facilitates the evaluation.

Substitution Method in Integration

The substitution method involves changing variables to simplify an integral. When integrating functions like cotangent powers, substituting u = cot(t) or using related expressions can transform the integral into a polynomial form, making it easier to solve.

Evaluate the integrals in Exercises 33–52.∫ 8 cot⁴(t) dt