Evaluate the integrals in Exercises 33–52.
∫ cot³(t) csc⁴(t) dt

Verified step by step guidance

1. Rewrite the integral \( \int \cot^{3}(t) \csc^{4}(t) \, dt \) by expressing powers of cotangent and cosecant in terms of sine and cosine, or by separating one factor to facilitate substitution. For example, write \( \cot^{3}(t) = \cot(t) \cdot \cot^{2}(t) \) and recall that \( \cot^{2}(t) = \csc^{2}(t) - 1 \).
2. Substitute \( \cot^{2}(t) = \csc^{2}(t) - 1 \) into the integral to rewrite it as \( \int \cot(t) (\csc^{2}(t) - 1) \csc^{4}(t) \, dt \), which simplifies to \( \int \cot(t) (\csc^{6}(t) - \csc^{4}(t)) \, dt \).
3. Split the integral into two separate integrals: \( \int \cot(t) \csc^{6}(t) \, dt - \int \cot(t) \csc^{4}(t) \, dt \). This allows you to handle each integral individually.
4. Use the substitution \( u = \csc(t) \), which implies \( du = -\csc(t) \cot(t) \, dt \). Rearrange to express \( \cot(t) \, dt \) in terms of \( du \) and \( u \), and rewrite each integral accordingly.
5. After substitution, integrate the resulting expressions in terms of \( u \) by applying the power rule for integrals. Finally, substitute back \( u = \csc(t) \) to express the answer in terms of \( t \).

Key Concepts

Trigonometric Identities

Trigonometric identities like cot²(t) + 1 = csc²(t) help simplify expressions involving powers of cotangent and cosecant. Recognizing and applying these identities allows rewriting the integral in a more manageable form for integration.

Integration of Powers of Trigonometric Functions

Integrating powers of trigonometric functions often requires reducing the powers using identities or substitution. Techniques include expressing higher powers in terms of lower powers or converting to sine and cosine to facilitate integration.

Substitution Method

The substitution method involves choosing a part of the integrand as a new variable to simplify the integral. For integrals involving cotangent and cosecant, substituting u = csc(t) or u = cot(t) can transform the integral into a basic form that is easier to evaluate.