sin(2x)sin(4x)sin(6x)

Question

pls give the answer of this in integration

Answer 1

Let X=sin(2x)sin(4x)sin(6x)=sin(2x)sin(4x)(sin(4x)cos(2x)+cos(4x)sin(2x))=

sin(2x).2sin(2x)cos(2x)(2sin(2x)cos^2(2x)+(1-2sin^2(2x))sin(2x))=

2sin^2(2x)cos(2x)(2sin(2x)(1-sin^2(2x))+sin(2x)-2sin^3(2x))=

2sin^2(2x)(3sin(2x)-4sin^3(2x))cos(2x)=sin^3(2x)(3-4sin^2(2x)).2cos(2x)dx.

We want ∫Xdx.

Let p=sin(2x), dp=2cos(2x)dx.

∫Xdx=∫p^3(3-4p^2)dp=∫(3p^3-4p^5)dp=(3/4)p^4-(2/3)p^6.

That is: (3/4)sin^4(2x)-(2/3)sin^6(2x)+C where C=constant of integration.