\[\begin{array}{l} \smallint se{c^{\bf{4}}}xta{n^{\bf{3}}}x{\rm{d}}x\\\\ \smallint {\sec ^2}{\rm{x}}{\tan ^3}{\rm{x}}{\sec ^2}{\rm{xdx}}\\ \smallint \left( {1 + {{\tan }^2}x} \right){\tan ^3}\;x{\sec ^2}x{\rm{d}}x\\ By\;Substitution\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Suppose \Rightarrow z = \tan x\\ \;\;\;\;\;\;\;\;\;\;\;\;\;dz = {\sec ^2}x\;\;dx\\\\ \smallint \left( {1 + {z^2}} \right){z^3}{\rm{d}}z\\ \smallint {z^5} + {z^3}{\rm{d}}z\\ \Rightarrow \frac{{{z^6}}}{6} + \frac{{{z^4}}}{4} + c\\ \Rightarrow \frac{{{{\tan }^6}x}}{6} + \frac{{{{\tan }^4}x}}{4} + c \end{array}\]
Tags:
integration