Evaluate the Integral integral of sin(3x)^2 with respect to x

Problem

(∫_^)(sin2(3*x)*d(x))

Solution

1. Apply the power-reduction identity to rewrite the integrand, using the formula sin2(θ)=(1−cos(2*θ))/2

2. Substitute θ=3*x into the identity to get sin2(3*x)=(1−cos(6*x))/2

3. Rewrite the integral by pulling out the constant factor of 1/2

(∫_^)(sin2(3*x)*d(x))=(∫_^)((1−cos(6*x))/2*d(x))

(∫_^)(sin2(3*x)*d(x))=1/2*(∫_^)((1−cos(6*x))*d(x))

4. Integrate term by term using the basic integration rules (∫_^)(1*d(x))=x and (∫_^)(cos(a*x)*d(x))=sin(a*x)/a

1/2*(∫_^)((1−cos(6*x))*d(x))=1/2*(x−sin(6*x)/6)+C

5. Distribute the constant 1/2 to simplify the expression.

1/2*x−sin(6*x)/12+C

Final Answer

(∫_^)(sin2(3*x)*d(x))=x/2−sin(6*x)/12+C

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