∫sin(5x/2)sin(x/2)dx is equal to: (where c is a constant of integration)

Correct option is C.
x+2sinx+sin2x+c
∫sin(5x/2)sin(x/2)dx = ∫2sin(5x/2)cos(x/2)2sin(x/2)cos(x/2)dx = ∫sin(3x)+sin(2x)sin(x)dx
=∫(3sinx−4sin³x)dx = ∫(3sinx−2(1−cos2x)+2cosx)dx
=∫(3−2(1−cos2x)+2cosx)dx = x+sin2x+2sinx+c