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
⇒
⇒
⇒
⇒ .