Integrate tan^6(x)

Answer 1

use tan^2x = sec^2x - 1

tan^6 = tan^4 (sec^2 - 1)
= tan^4 sec^2 - tan^4
= tan^4 sec^2 - tan^2 sec^2 + tan^2
= tan^4 sec^2 - tan^2 sec^2 + sec^2 - 1

now d(tanx) = sec^2x dx, so what you have is

u^4 du - u^2 du + du - dx
= 1/5 u^5 - 1/3 u^3 + u - x

= 1/5 tan^5x - 1/3 tan^3x + tanx - x + C