证明

\[
\int\sec xdx=\int\frac{1}{\cos x}dx=\ln |\sec x+\tan x|+C.
\]

\[
\int\csc xdx=\int\frac{1}{\sin x}dx=\frac{1}{2}\ln\biggl|\frac{1-\cos x}{1+\cos x}\biggr|+C=\ln\biggl|\tan\frac{x}{2}\biggr|+C.
\]

并求 $\int \sec^2 xdx$, $\int \sec^3 xdx$

[总结]

\[
\begin{aligned}
\int\sec xdx&=\ln|\sec x+\tan x|+C\\
\int\sec^2 xdx&=\tan x+C\\
\int\sec^3 xdx&=\frac{1}{2}\ln|\sec x+\tan x|+\frac{1}{2}\sec x\cdot\tan x+C
\end{aligned}
\]