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Use substitution. Let u = ln(3x).

Use integration by parts. Let f = sin(2x) and dg = exdx.
Then df = 2cos(2x)dx and g = ex.

Use integration by parts on the integral in Step 2.
Let f = cos(2x) and dg = exdx.
Then df = –2sin(2x)dx and g = ex.

Use partial fraction decomposition.

Rewrite the denominator of the integrand using Steps 1 and 2.
You should recognize the integrand as a form of the derivative of a special function.

Use algebra to get the original integral expression by itself.