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.

Integration Techniques

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.

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.

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