∫sin(3x + a).sin(x -a) dx
let......... I =∫sin(3x + a).sin(x -a) dx
let...... u = sin(3x - a)............ du/dx = 3cos(3x + a)
dv =sin(x - a) dx................ v = -cos(x - a)
I = -cos(x -a).sin(3x - a) + 3∫cos(3x + a)cos(x -a) dx
Again
let........ u = cos(3x + a)................ du/dx = -3sin(3x + a)
dv = cos(x - a) dx................. v = sin(x - a)
∫cos(3x + a)cos(x -a) dx = cos(3x + a)sin(x - a) +3∫sin(3x + a)sin(x - a)
=cos(3x + a)sin(x - a) + 3I
I = -cos(x - a)sin(3x + a) + 3{cos(3x + a)sin(x - a)} + 9I
I = 1/8{cos(x - a)sin(3x + a) -3cos(3x + a)sin(x - a)}
=by expressing in the form of sinP + sinQ = 2sin1/2(P + Q)cos1/2(P - Q) and
sinP - sinQ =2cos1/2(P + Q)sin(P -Q)
∫sin(3x + a).sin(x -a) dx=1/4sin(2x + 2a) - 1/8sin4x + c