intefrated the function f(x)on Rif f(x) =sinx.sin2x.sin3x.dx

Dear Student

LetI=∫sinx.sin2x.sin3xdx
=(1/2) ∫2sinx.sin2x.sin3xdx
= (1/2) ​∫sinx.(2sin2x.sin3x)dx
We know that[2sinA.sinB=cos(A−B)−cos(A+B)]
So,= (1/2)∫sinx.(cosx−cos5x)dx
=(1/2)(1/2)∫2sinx.cosxdx−2×21​∫2sinx.cos5xdx
= (1/4) ​∫sin2xdx−(1/4)​∫(sin6x−sin4x)dx
=− (1/8)cos2x​+(1/24)cos6x​−(1/16)cos4x​+c

Thanks