[M@gpie]

Let : \( u =\sec x \; \text{and} \; dv=\sec^2 x \ dx\)
then \( v=\int \sec^2 x \ dx = \tan x \)
ทำการอินทิเกรตทีละส่วน
\[ \begin{array}{rcl} \int \sec^3 x \ dx &= & \sec x \tan x - \int \tan x \ d(\sec x) \\
& = & \sec x \tan x - \int \tan^2 x \sec x \ dx \\
& = & \sec x \tan x - \int (1+ \tan^2 x) \sec x \ dx \\
& = & \sec x \tan x - \int \sec^3 x \ dx + \int \sec x \ dx \\ 2 \int \sec^3 x \ dx & = & \sec x \tan x - ln \mid \sec x +\tan x \mid + C \\
\int \sec^3 x \ dx & = & \frac{1}{2} \sec x \tan x - \frac{1}{2} \ln \mid \sec x +\tan x \mid + C
\end{array} \]