Evaluate the Integral

Problem

(∫_^)(sin2(θ)*d(θ))

Solution

1. Identify the trigonometric identity for sin2(θ) to reduce the power, as the integral of a squared sine function is not a basic form.

2. Apply the identity for the power-reduction formula:

sin2(θ)=(1−cos(2*θ))/2

3. Substitute the identity into the integral:

(∫_^)((1−cos(2*θ))/2*d(θ))

4. Factor out the constant 1/2 from the integral:

1/2*(∫_^)((1−cos(2*θ))*d(θ))

5. Integrate the terms individually with respect to θ

1/2*(θ−sin(2*θ)/2)+C

6. Distribute the constant to simplify the expression:

θ/2−sin(2*θ)/4+C

Final Answer

(∫_^)(sin2(θ)*d(θ))=θ/2−sin(2*θ)/4+C

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