\[\begin{array}{l} \smallint ta{n^{\bf{3}}}x{\rm{d}}x\\\\ \smallint tanxta{n^2}x{\rm{d}}x\\ \smallint \tan x\left( {{{\sec }^2}x - 1} \right){\rm{d}}x\\ \smallint tanxse{c^2}x{\rm{d}}x - \smallint tanx{\rm{d}}x\\ By\;Substitution\\ Suppose \Rightarrow z = \sec x\;\\ \;\;\;\;\;\;\;\;\;\;dz = \sec x\;\tan x\;dx\\ \smallint z{\rm{d}}z + \ln \left| {\cos x} \right| + c\\ \Rightarrow \frac{{{z^2}}}{2} + \;\ln \left| {\cos x} \right| + c\\ \Rightarrow \frac{{{{\sec }^2}x}}{2} + \ln \left| {\cos x} \right| + c \end{array}\]
Tags:
integration