How do you integrate sin(3x)cos(5x)?

STEP 1 Cannot integrate directly in this form, therefore use a trigonometric identity. Identity: sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2) STEP 2 (A+B)/2 = 3x             A + B = 6x   (1) (A-B)/2 = 5x              A - B = 10x     (2) STEP 3 Solve simultaneous equations (1) and (2) (1) + (2)      2A = 16x    A = 8x Substitute A into (1)      B = -2x STEP 4 Substitute values for A and B into the trigonometric identity. sin(3x)cos(5x) = (1/2)(sin(8x) + sin(-2x)) STEP 5 Now integrate. {Note: integral of sine is negative cosine} = (1/2)(integral sign)[sin(6x) + sin(-2x)] dx = (1/2)[(1/8)(-cos(8x)) + (1/2)cos(-2x)] + C = (-1/16)cos(8x) + (1/4)cos(-2x) + C //