text: # 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