Evaluate the Integral

Problem

(∫_^)(cos2(θ)*d(θ))

Solution

1. Identify the power-reduction identity for the squared trigonometric function to make it integrable.

2. Apply the formula cos2(θ)=(1+cos(2*θ))/2 to rewrite the integrand.

(∫_^)(cos2(θ)*d(θ))=(∫_^)((1+cos(2*θ))/2*d(θ))

3. Factor out the constant 1/2 from the integral.

1/2*(∫_^)((1+cos(2*θ))*d(θ))

4. Integrate each term separately with respect to θ

1/2*(θ+sin(2*θ)/2)+C

5. Distribute the constant to simplify the final expression.

θ/2+sin(2*θ)/4+C

Final Answer

(∫_^)(cos2(θ)*d(θ))=θ/2+sin(2*θ)/4+C

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