2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
32. ∫ csc²(6x) cot(6x) dx

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Step 1: Recognize the integral ∫ csc²(6x) cot(6x) dx as a product of trigonometric functions. Notice that csc²(6x) is the derivative of cot(6x). This suggests a substitution method might be effective.

Step 2: Let u = cot(6x). Then, compute the derivative of u with respect to x: du/dx = -6 csc²(6x). Rearrange to express dx in terms of du: dx = -du / (6 csc²(6x)).

Step 3: Substitute u = cot(6x) and dx = -du / (6 csc²(6x)) into the integral. The csc²(6x) term in the original integral will cancel out, leaving ∫ u (-du / 6).

Step 4: Simplify the integral to ∫ (-u / 6) du. Factor out the constant -1/6 to get (-1/6) ∫ u du.

Step 5: Integrate u with respect to u. The integral of u is u²/2. Multiply by the constant -1/6 to get (-1/6)(u²/2). Substitute back u = cot(6x) to express the result in terms of x.

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Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric identities. Understanding these methods is crucial for solving integrals that may not be straightforward, as they allow for simplification and manipulation of the integrand.

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Trigonometric Functions

Trigonometric functions, such as sine, cosine, and their derivatives, play a significant role in calculus. In this problem, csc²(6x) and cot(6x) are trigonometric functions that can be integrated using their known derivatives. Recognizing the relationships between these functions is essential for applying the appropriate integration techniques.

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Substitution Method

The substitution method is a powerful technique in integration that involves changing the variable of integration to simplify the integral. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.

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