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

Let . Then , so . Rewrite using and .
Let . Find .
Rewrite.
Divide by .
Rewrite the problem using and .
Combine and .
Since is constant with respect to , move out of the integral.
Use the halfangle formula to rewrite as .
Since is constant with respect to , move out of the integral.
Simplify.
Multiply and .
Multiply by .
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
Since is constant with respect to , move out of the integral.
Let . Then , so . Rewrite using and .
Let . Find .
Rewrite.
Divide by .
Rewrite the problem using and .
Combine and .
Since is constant with respect to , move out of the integral.
The integral of with respect to is .
Simplify.
Substitute back in for each integration substitution variable.
Replace all occurrences of with .
Replace all occurrences of with .
Replace all occurrences of with .
Simplify each term.
Multiply by .
Combine and .
Apply the distributive property.
Cancel the common factor of .
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Multiply .
Multiply and .
Multiply by .
Reorder terms.
Evaluate integral of sin(3x)^2 with respect to x

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