2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
51. ∫ (from 0 to π/4) sin⁵(4θ) dθ

Verified step by step guidance

1. Step 1: Recognize that the integral involves a power of sine, specifically sin⁵(4θ). To simplify, use the power-reduction formula for sine: sin²(x) = (1 - cos(2x))/2. This will help break down the higher power of sine into manageable terms.
2. Step 2: Rewrite sin⁵(4θ) as (sin²(4θ))² * sin(4θ). Substitute the power-reduction formula for sin²(4θ), which becomes ((1 - cos(8θ))/2)² * sin(4θ). Expand this expression to prepare for integration.
3. Step 3: Expand ((1 - cos(8θ))/2)², which results in (1/4)(1 - 2cos(8θ) + cos²(8θ)). Replace cos²(8θ) using the power-reduction formula: cos²(x) = (1 + cos(2x))/2. This substitution simplifies the expression further.
4. Step 4: After substitution, the integral becomes a sum of terms involving sin(4θ), cos(8θ), and cos(16θ). Split the integral into separate parts for each term, and use standard integration techniques for trigonometric functions. For example, ∫sin(kθ)dθ = -(1/k)cos(kθ) and ∫cos(kθ)dθ = (1/k)sin(kθ).
5. Step 5: Evaluate each term of the integral from 0 to π/4. Substitute the limits of integration into the antiderivative expressions obtained in the previous step. Simplify the results to complete the evaluation of the 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 crucial for evaluating complex integrals, especially those involving powers of trigonometric functions, as seen in the given integral.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of triangles, and they can simplify the integration process. For example, identities like sin²(θ) + cos²(θ) = 1 can help rewrite higher powers of sine and cosine functions. Recognizing and applying these identities is essential for transforming the integrand into a more manageable form.

Definite Integrals

A definite integral calculates the area under a curve between two specified limits. It is represented as ∫ from a to b f(x) dx, where 'a' and 'b' are the bounds of integration. Understanding how to evaluate definite integrals, including applying the Fundamental Theorem of Calculus, is necessary for solving the integral in the question.