Question

Solve the following integral, show all steps.

Answer 1

Use integration by parts:
$u = x ............ dv = sec^2 x \\ \\ du = dx ....... v = tan x \\ \\ =x tan( x) - \int tan (x) dx$

For the integral of tangent, multiply by sec on top and bottom.
$= x tan(x) - \int \frac{sec (x) tan(x)}{sec(x)} dx$

Use u-substitution:
$u = sec(x) \\ \\ du = sec(x) tan(x) dx \\ \\ =x tan(x) - \int \frac{du}{u} \\ \\ =x tan(x) - ln |u| + C \\ \\ = x tan(x) - ln|sec(x)| + C$

Answer can also be written as
$x tan(x) + ln|cos(x)| + C$