## anonymous 5 years ago integrate tan to the power 5X

1. anonymous

$\tan(x)^{5x}$ Is that the expression?

2. anonymous

variable in the exponent, consider logarthmic manipulation: integrate 5x ln tan theta

3. anonymous

Yeah, you didn't say tan of what

4. 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.

5. anonymous

Where is oneprince?

6. anonymous

The prince has stepped in his concubine to satisfy one of his maidens

7. anonymous

INweton integrate it.that is the correct question

8. 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 :@.

9. anonymous

You simply let tan x be u and integrate u^5

10. anonymous

INewton continue

11. 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.

12. anonymous

Chaguanas, that doesn't work well because you'll end up with a factor $$cos^2x$$ from the substitution of dx.

13. anonymous

I think it does work (sec^2x = 1 + tan^2x) but it's not MUCH nicer. Again, pretty tired so may be wrong.

14. anonymous

Actually, looking at what comes out I'd wager not nicer full stop.

15. anonymous

Alternative method: tables: integration tan^u du=(1/(n-1)) tan^(n-1) u - integral tan^(n-2) u du