What is the integral of sin(3x) cos(5x)?

Using trig formulas we have sin(5x+3x)= sin(5x)cos(3x)+cos(5x)sin(3x) and sin(5x-3x)= sin(5x)cos(3x) - cos(5x)sin(3x). Hence, sin(5x+3x) - sin(5x-3x) = sin(8x)-sin(2x) = 2cos(5x)sin(3x). This implies that cos(5x)sin(3x)=(sin(8x)-sin(2x))/2 which we can easily integrated using the reverse chain-rule to get:

(-cos(8x)/8 + cos(2x)/2)/2 + C, simplifying further we get (4cos(2x) - cos(8x))/2 + C.