Question

9–61. Trigonometric integrals Evaluate the following integrals.32. ∫ cot⁵(3x) dx

Accepted Answer

Step 1: Recognize that the integral involves a power of cotangent. To simplify, use the trigonometric identity $$ \cot^2(x) = \csc^2(x) - 1  to $$express higher powers of $$ \cot(x)  in $$terms of $$ \csc(x) . $$Rewrite $$ \cot^5(3x)  as$$ $$ \cot^3(3x) \cdot \cot^2(3x) . $$Step 2: Substitute $$ \cot^2(3x) $$ using the identity $$ \cot^2(3x) = \csc^2(3x) - 1 . $$This transforms the integral into $$ \int \cot^3(3x) \cdot (\csc^2(3x) - 1) \, dx . $$Step 3: Break the integral into two parts: $$ \int \cot^3(3x) \cdot \csc^2(3x) \, dx $$ and $$ -\int \cot^3(3x) \, dx . $$Focus on the first term by using substitution. Let $$ u = \cot(3x) $$, so $$ du = -3 \csc^2(3x) \, dx . $$Rewrite the integral in terms of $$ u . $$Step 4: After substitution, the first term becomes $$ -\frac{1}{3} \int u^3 \, du . $$Solve this integral using the power rule for integration: $$ \int u^n \, du = \frac{u^{n+1}}{n+1} . $$For the second term, rewrite $$ \cot^3(3x)  as$$ $$ \cot(3x) \cdot \cot^2(3x) $$, and use the identity $$ \cot^2(3x) = \csc^2(3x) - 1 $$ again. Step 5: Combine the results of the two integrals, back-substitute $$ u = \cot(3x) $$, and simplify the expression. Ensure the final answer includes the constant of integration $$ C .$$

9–61. Trigonometric integrals Evaluate the following integrals.
32. ∫ cot⁵(3x) dx

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주요 개념

Trigonometric Functions

Integration Techniques

Power Reduction and Trigonometric Identities