# How to evaluate the Trigonometric Integrals : ∫ sin^2(1/3 θ) dθ , the upper limit is 2π and the lower limit is 0 ?

Use the power reducing formula.

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To evaluate the integral ∫ sin²(1/3 θ) dθ from 0 to 2π, we can use the trigonometric identity sin²(θ) = 1/2 - 1/2 cos(2θ):

∫ sin²(1/3 θ) dθ = ∫ (1/2 - 1/2 cos(2(1/3 θ))) dθ = ∫ (1/2 - 1/2 cos(2/3 θ)) dθ = (1/2)θ - (1/4)sin(2/3 θ) + C

Evaluate this expression from 0 to 2π:

[(1/2)(2π) - (1/4)sin(2/3 (2π))] - [(1/2)(0) - (1/4)sin(2/3 (0))] = π - 0 = π

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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