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

∫₋π^π cos² 𝓍 d𝓍

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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