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In my university website there is a PDF with some integrals and I found this formula for the integral of the secant function to any power: $$\int\sec^nx \mathrm dx=\frac{\tan x \sec^{n-2}x}{n-1}+\frac{n-2}{n-1}\int\sec^{n-2}x\mathrm dx \,\,\,\,\,\,\, (n\neq 1).$$

I searched for it on the internet, but I couldn't find anything. I know that it works for $n=2$ and for $n=3$, but I'm not sure if it works for any other power. Something like $n\in\mathbb N-\{1\}$ is never specified, but I assume that since $n$ is usually used for natural numbers.

Is that formula correct? And, if it is indeed correct, how do you prove it?

Ciro
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  • I have no idea (off the top of my head) if the formula is correct, but it looks like IBP and a consequence of the Pythagorean identity to me. – Xander Henderson Feb 07 '22 at 14:36
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    See here – anankElpis Feb 07 '22 at 14:42
  • Integration by parts $\sec^2 xdx=d(tan x)$ – Vasili Feb 07 '22 at 14:47
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    Look up reduction formulae. – r00r Feb 07 '22 at 15:25
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    The question seems to have been thoroughly answered even though the "duplicate" question wasn't phrased quite the same way. In particular there is a reasonably complete proof there, in which it doesn't even seem necessary for $n$ to be an integer. In fact it's only the division by $n-1$ at the very end that requires $n\neq 1.$ But the question could be reopened if there is some really new issue that the answers did not cover. – David K Feb 07 '22 at 17:57
  • @DavidK Thank you for finding that duplicate target. I had intended to revisit this question a little later in the day, when I am done with students. I am glad that you were able to find a good answer for the asker here in less time. – Xander Henderson Feb 07 '22 at 19:36
  • @XanderHenderson Credit Stefan for the find in the second comment. I merely decided it was good enough for a duplicate. – David K Feb 08 '22 at 02:31
  • Indeed. Thank you, @StefanAlbrecht. – Xander Henderson Feb 08 '22 at 12:42

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