Calculus: Proving a reduction formula?

Question

How to prove the reduction formula Int((tan^n)x dx = (1/n-1)tan^(n-1)x) - Int (tan^n-2 x dx)

I'm so lost.

Answer 1

∫(tan x)^n dx

First, isolate a factor of tan²:

∫(tan x)^(n-2) tan² x dx

Now use the fact that tan² x=sec² x - 1

∫(tan x)^(n-2) (sec² x - 1) dx

Distribute, and break this up into two integrals:

∫(tan x)^(n-2) sec² x dx - ∫(tan x)^(n-2) dx

Make a substitution in the left integral: u=tan x, du=sec² x dx

∫u^(n-2) du - ∫(tan x)^(n-2) dx

Integrate using the power rule:

u^(n-1)/(n-1) - ∫(tan x)^(n-2) dx

Finally, resubstitute tan x:

(tan x)^(n-1)/(n-1) - ∫(tan x)^(n-2) dx

And we are done.