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

∫₋π^π cos² 𝓍 d𝓍

Guida verificata passo dopo passo

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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