9–61. Trigonometric integrals Evaluate the following integrals.
11. ∫ sin²(3x) dx

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Step 1: Recognize that the integral involves a squared trigonometric function, sin²(3x). To simplify, use the trigonometric identity: sin²(u) = (1 - cos(2u)) / 2. Substitute u = 3x into the identity.

Step 2: Rewrite the integral using the identity: ∫ sin²(3x) dx = ∫ (1 - cos(6x)) / 2 dx. This simplifies the integral into two separate terms.

Step 3: Split the integral into two parts: ∫ sin²(3x) dx = (1/2) ∫ 1 dx - (1/2) ∫ cos(6x) dx. Now, handle each term individually.

Step 4: Evaluate the first term, (1/2) ∫ 1 dx, which is straightforward and results in (1/2)x. For the second term, (1/2) ∫ cos(6x) dx, use the substitution method. Let u = 6x, then du = 6 dx, and rewrite the integral accordingly.

Step 5: After substitution, integrate cos(u) to get sin(u). Substitute back u = 6x into the result. Combine the results from both terms to express the final solution, adding the constant of integration C.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. A key identity for integrating sine squared functions is the Pythagorean identity, which states that sin²(x) = (1 - cos(2x))/2. This identity simplifies the integration process by transforming the sine function into a form that is easier to integrate.

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Integration techniques are methods used to find the integral of a function. For the integral of sin²(3x), one common technique is substitution, where we can let u = 3x, making du = 3dx. This allows us to rewrite the integral in terms of u, simplifying the integration process and making it more manageable.

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Definite vs. Indefinite Integrals

Definite and indefinite integrals are two types of integrals in calculus. An indefinite integral, like ∫ sin²(3x) dx, represents a family of functions and includes a constant of integration (C). In contrast, a definite integral calculates the area under the curve between two specific limits. Understanding the difference is crucial for correctly interpreting the results of integration.

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