How do you integrate #tan^5(x)#?

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Answer 1 · 2017-10-31T16:49:05

Recursively use the reduction formula:

#inttan^n(x)dx=1/(n-1)tan^(n-1)(x)-inttan^(n-2)(x)dx#

Until #n = 1#, then we know this one.

Starting with #n = 5#:

#inttan^5(x)dx=1/4tan^4(x)-inttan^3(x)dx#

Now #n = 3#:

#inttan^5(x)dx=1/4tan^4(x)-1/2tan^2(x)+inttan(x)dx#

We know the last integral:

#inttan^5(x)dx=1/4tan^4(x)-1/2tan^2(x)+ln|sec(x)|+ C#