Evaluate the Integral integral of sin(x)^2 with respect to x

Problem

(∫_^)(sin2(x)*d(x))

Solution

1. Identify the trigonometric identity for sin2(x) to reduce the power, as it is not a basic integral.

2. Apply the identity sin2(x)=(1−cos(2*x))/2 to rewrite the integrand.

(∫_^)(sin2(x)*d(x))=(∫_^)((1−cos(2*x))/2*d(x))

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

1/2*(∫_^)((1−cos(2*x))*d(x))

4. Integrate the terms individually using the rule (∫_^)(cos(a*x)*d(x))=sin(a*x)/a

1/2*(x−sin(2*x)/2)+C

5. Distribute the constant to simplify the final expression.

x/2−sin(2*x)/4+C

Final Answer

(∫_^)(sin2(x)*d(x))=x/2−sin(2*x)/4+C

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