Question

integrate tan to the power 5X

Answer 1

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

Answer 2

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

Answer 3

Yeah, you didn't say tan of what

Answer 4

I'd guess (in the absence of any variable except x) that it is:

\[\tan^{5}x \]

But of course, I could be wrong.

Answer 5

Where is oneprince?

Answer 6

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

Answer 7

INweton integrate it.that is the correct question

Answer 8

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 :@.

Answer 9

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

Answer 10

INewton continue

Answer 11

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.

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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