Evaluate the following integrals:

∫ tan 2x tan 3x tan 5x dx

We know tan5x = tan(2x + 3x)

tan(A + B) =

∴ tan(2x + 3x) =

∴ tan(5x) =

⇒ tan(5x)(1 - tan2x.tan3x) = tan(2x) + tan(3x)

⇒ tan(5x) - tan2x.tan3x.tan5x = tan(2x) + tan(3x)

⇒ tan(5x) - tan(2x) - tan(3x) = tan2x.tan3x.tan5x

Substituting the above result in given equation we get

⇒

⇒

⇒

⇒ .