# tan reduction formula

• Jul 25th 2010, 02:01 PM
Tclack
tan reduction formula
So the proof for the reduction formula for [tan(x)]^n I found here, and it makes sense

But I did this and it is very close, but not the same formula, what's wrong?

$\displaystyle \int tan^nx dx = \int tan^n^-^2x *tan^2x dx$
$\displaystyle =\int tan^n^-^2x(sec^2x -1) dx$
$\displaystyle = \int tan^n^-^2x*sec^2x dx - \int tan^n^-^2x dx$

here I did an integration by parts of the first integral above:

$\displaystyle u= tan^n^-^2x,,,,,,,,,,,,,,,dv=sec^2xdx$
$\displaystyle du=(n-2)tan^n^-^3xdx,,,,,,v=tanx$

$\displaystyle uv-\int vdu$

$\displaystyle tan^n^-^1x-(n-2)\int tan^n^-^2xdx - \int tan^n^-^2xdx$

so I get
$\displaystyle \int tan^nx = tan^n^-^1x-(n-1)\int tan^n^-^2xdx$ Where's did I go wrong?
• Jul 25th 2010, 02:23 PM
drumist
The derivative of $\displaystyle \tan^{n-2} x$ is not $\displaystyle (n-2) \tan^{n-3} x$. Do you see why?
• Jul 25th 2010, 03:00 PM
Tclack
Oh yeah!
Yeah, the Chain Rule. Thanks for the backup