Find the Integral cos(theta)^2

Problem

(∫_^)(cos2(θ)*d(θ))

Solution

1. Apply the power-reduction identity to rewrite the squared trigonometric function into a linear form.

cos2(θ)=(1+cos(2*θ))/2

2. Substitute the identity into the integral and factor out the constant coefficient.

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

3. Distribute the integral across the sum of the two terms.

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

4. Evaluate the integrals using basic integration rules and the reverse chain rule for the cosine term.

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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