Use integration by parts:
[tex]u = x ............ dv = sec^2 x \\ \\ du = dx ....... v = tan x \\ \\ =x tan( x) - \int tan (x) dx[/tex]
For the integral of tangent, multiply by sec on top and bottom.
[tex]= x tan(x) - \int \frac{sec (x) tan(x)}{sec(x)} dx[/tex]
Use u-substitution:
[tex]u = sec(x) \\ \\ du = sec(x) tan(x) dx
\\ \\ =x tan(x) - \int \frac{du}{u} \\ \\ =x tan(x) - ln |u| + C \\ \\ = x tan(x) - ln|sec(x)| + C[/tex]
Answer can also be written as
[tex]x tan(x) + ln|cos(x)| + C[/tex]
