2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
48. ∫ sin(3x) cos⁶(3x) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves a product of trigonometric functions, specifically sin(3x) and cos⁶(3x). To simplify, consider using a substitution method. Let u = cos(3x), which implies that du = -3sin(3x) dx.
2. Step 2: Rewrite the integral in terms of u. Substitute sin(3x) dx with -du/3 and cos⁶(3x) with u⁶. The integral becomes: ∫ sin(3x) cos⁶(3x) dx = -1/3 ∫ u⁶ du.
3. Step 3: Apply the power rule for integration to evaluate ∫ u⁶ du. Recall that the power rule states: ∫ uⁿ du = uⁿ⁺¹ / (n+1) + C, where n ≠ -1.
4. Step 4: Substitute back u = cos(3x) into the result obtained from the integration step to express the solution in terms of x.
5. Step 5: Add the constant of integration (C) to complete the solution, as this is an indefinite integral.

Key Concepts

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 essential for evaluating complex integrals, as they allow for simplification and manipulation of the integrand to facilitate integration.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of triangles, and they are crucial in simplifying integrals involving trigonometric functions. For example, the product-to-sum identities can transform products of sine and cosine into sums, making integration more manageable. Familiarity with these identities is vital for solving integrals like ∫ sin(3x) cos⁶(3x) dx.

Substitution Method

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