\[\begin{array}{l} \smallint se{c^{\bf{3}}}x{\rm{d}}x\\\\ \;\smallint secxse{c^2}x{\rm{d}}x\\ By\;Parts\\ Let \Rightarrow u = \sec x\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;dv = se{c^2}x\\ \;du = \tan x\sec xdx\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;v = \tan x\\ \sec x\tan x - \smallint {\tan ^2}x\sec x{\rm{d}}x\\ \sec x\tan x - \smallint \sec x\left( {{{\sec }^2}x - 1} \right){\rm{d}}x\\ \Rightarrow \smallint {\sec ^3}x{\rm{d}}x = \sec x\tan x - \smallint {\sec ^3}x{\rm{d}}x + \smallint \sec x{\rm{d}}x\\ \Rightarrow 2\smallint {\sec ^3}{\rm{d}}x = \sec x\tan x + \ln \left| {\sec x + \tan x} \right| + c\\ \Rightarrow \smallint {\sec ^3}{\rm{d}}x = \frac{1}{2}(\sec x\tan x + \ln \left| {\sec x + \tan x} \right|) + c \end{array}\]
Tags:
integration