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

∫₋π^π cos² 𝓍 d𝓍

검증된 단계별 안내

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

Step 2: Rewrite the integral using the identity: ∫₋π^π cos²(𝓍) d𝓍 = ∫₋π^π (1 + cos(2𝓍)) / 2 d𝓍.

Step 3: Split the integral into two separate integrals: ∫₋π^π (1/2) d𝓍 + ∫₋π^π (cos(2𝓍)/2) d𝓍.

Step 4: Evaluate the first integral ∫₋π^π (1/2) d𝓍. This is a constant term, so it simplifies to (1/2) * ∫₋π^π d𝓍, which is the length of the interval multiplied by 1/2.

Step 5: Evaluate the second integral ∫₋π^π (cos(2𝓍)/2) d𝓍. Since cos(2𝓍) is an even function and the interval is symmetric about zero, the integral of cos(2𝓍) over [-π, π] is zero. Combine the results from both integrals to complete the solution.

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주요 개념

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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 sin²(x) and cos²(x), applying the half-angle identities can transform the integrals into more manageable forms.

Definite Integrals

Definite integrals represent the signed area under a curve between two specified limits. The notation ∫_a^b f(x) dx indicates 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 connects differentiation and integration.

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals. ∫₋π^π cos² 𝓍 d𝓍