Asked by Shayna
Integrate tan^6(x)
Answers
Answered by
Steve
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
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
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