Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.

∫₀^π/⁴ cos² 8θ dθ

검증된 단계별 안내

Step 1: Recognize that the integral involves cos²(8θ). To simplify this, use the trigonometric identity cos²(x) = (1 + cos(2x)) / 2.

Step 2: Substitute the identity into the integral. The integral becomes ∫₀^(π/4) [(1 + cos(16θ)) / 2] dθ.

Step 3: Split the integral into two separate integrals: (1/2) ∫₀^(π/4) 1 dθ + (1/2) ∫₀^(π/4) cos(16θ) dθ.

Step 4: Evaluate the first integral, (1/2) ∫₀^(π/4) 1 dθ, which is straightforward as it represents the area under a constant function. For the second integral, (1/2) ∫₀^(π/4) cos(16θ) dθ, use the formula for the integral of cos(kx), which is (1/k) sin(kx).

Step 5: Apply the limits of integration (0 to π/4) to both parts of the integral. For the first part, calculate the result of (1/2) θ evaluated at the limits. For the second part, calculate (1/2) * (1/16) * sin(16θ) evaluated at the limits.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, such as sin²(x) + cos²(x) = 1, and double angle formulas. These identities are essential for simplifying integrals involving sin²(x) and cos²(x), allowing for easier evaluation.

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and using trigonometric identities to simplify the integrand. For integrals involving cos²(θ), applying the identity cos²(θ) = (1 + cos(2θ))/2 can transform the integral into a more manageable form.

Definite Integrals

Definite integrals calculate the area under a curve between two specified limits. The notation ∫ₐᵇ f(x) dx represents the integral of f(x) from a to b. Evaluating definite integrals often involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus, which states that the definite integral can be computed by evaluating the antiderivative at the upper and lower limits.

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals. ∫₀^π/⁴ cos² 8θ dθ