Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 20

intxtan^2xdx
Rewrite the integrand using the identity tan^2x=sec^2x-1
intxtan^2xdx=intx(sec^2x-1)dx
=intxsec^2xdx-intxdx
Now let's evaluate intxsec^2xdx using integration by parts,
intxsec^2xdx=x*intsec^2xdx-int(d/dx(x)intsec^2(x))dx
=xtan(x)-int(1*tan(x))dx
=xtan(x)-int(sin(x)/cos(x))dx
Substitute cos(x)=t
-sin(x)dx=dt
int(sin(x)/cos(x))dx=int-dt/t
=-ln|t|
substitute back t=cos(x),
=-ln|cos(x)|
intxsec^2xdx=xtan(x)-(-ln|cos(x)|)
=xtan(x)+ln|cos(x)|
intxtan^2(x)dx=xtan(x)+ln|cos(x)|-intxdx
=xtan(x)+ln|cos(x)|-x^2/2+C
C is a constant