\displaystyle 太久 没有在知乎写回答了，今天拿到电脑的第一时间就想着水一篇文章，所以我们来水一下关于 \int \sec^nxdx的递推公式吧\displaystyle I_n=\int \sec^nxdx\

\displaystyle =\int \sec^{n-2}xd(\tan x)

\displaystyle =\tan x\sec^{n-2}x-(n-2)\int \tan xd(\sec^nx)

\displaystyle =\tan x\sec^{n-2}x-(n-2)\int \sin^2x\sec^nxdx

\displaystyle =\tan x\sec^{n-2}x-(n-2)\int \tan^2\sec^{n-2}xdx

\displaystyle =\tan x\sec^{n-2}x-(n-2)\int (\sec^2x-1)\sec^{n-2}xdx

\displaystyle =\tan x\sec^{n-2}x-(n-2)\int \sec^nxdx-(n-2)\int \sec^{n-2}xdx

\displaystyle I_n+(n-2)\int \sec^nxdx=\tan x\sec^{n-2}x-(n-2)\int \sec^{n-2}xdx

\displaystyle (n-1)\int \sec^nxdx=\tan x\sec^{n-2}x-(n-2)\int \sec^{n-2}xdx

\displaystyle (n-1)I_n=\tan x\sec^{n-2}x-(n-2)\int \sec^{n-2}xdx

\displaystyle \color{blue}{(n-1)I_n=\tan x\sec^{n-2}x-(n-2)I_{n-2}}