MHB Can You Derive a Reduction Formula for the Integral $\int \sec^nx dx$?

[lfdahl]

Derive a reduction formula for the integral:

$\int \sec^nx dx, \;\;\; n \ge 2.$

- without any help from an online integrator.

 

[lfdahl]

Suggested solution:

Spoiler

\[\int \sec^nxdx \\\\= \int \sec^2x\sec^{n-2}xdx = \tan x \sec^{n-2}x-(n-2)\int \tan x \sec^{n-3}x \sec^{2}x \sin xdx \\\\= \tan x \sec^{n-2}x-(n-2)\int \sin^2 x\sec^nxdx \\\\= \tan x \sec^{n-2}x-(n-2)\int (1-\cos^2x)\sec^nxdx \\\\= \tan x \sec^{n-2}x-(n-2)\int \sec^nxdx+(n-2)\int \sec^{n-2}xdx \\\\ \Rightarrow \int \sec^nxdx = \frac{\tan x \sec^{n-2}x}{n-1}+\frac{(n-2)}{(n-1)}\int \sec^{n-2}xdx\]