anonymous
  • anonymous
integrate tan to the power 5X
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
\[\tan(x)^{5x}\] Is that the expression?
anonymous
  • anonymous
variable in the exponent, consider logarthmic manipulation: integrate 5x ln tan theta
anonymous
  • anonymous
Yeah, you didn't say tan of what

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anonymous
  • anonymous
I'd guess (in the absence of any variable except x) that it is: \[\tan^{5}x \] But of course, I could be wrong.
anonymous
  • anonymous
Where is oneprince?
anonymous
  • anonymous
The prince has stepped in his concubine to satisfy one of his maidens
anonymous
  • anonymous
INweton integrate it.that is the correct question
anonymous
  • anonymous
Re-write it as \[\tan x(\sec^2x -1)(\sec^2x -1) \] Expand and do it term-by-term. Reduction formulae seems unnecessary, and I can;t see anything quicker right now :@.
anonymous
  • anonymous
You simply let tan x be u and integrate u^5
anonymous
  • anonymous
INewton continue
anonymous
  • anonymous
OK, I'll do a little more. That expand to \[\tan x \sec^4x - 2\tan x\ \sec^2x + \tan x \] And note that \[\frac{\mathbb{d}}{\mathbb{d}x}\frac{1}{n}\ \sec^nx =\tan x\ \sec^nx \] Again, probably something far quicker.
anonymous
  • anonymous
Chaguanas, that doesn't work well because you'll end up with a factor \(cos^2x\) from the substitution of dx.
anonymous
  • anonymous
I think it does work (sec^2x = 1 + tan^2x) but it's not MUCH nicer. Again, pretty tired so may be wrong.
anonymous
  • anonymous
Actually, looking at what comes out I'd wager not nicer full stop.
anonymous
  • anonymous
Alternative method: tables: integration tan^u du=(1/(n-1)) tan^(n-1) u - integral tan^(n-2) u du

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