Integrate from pi/4 to pi for sec x (sec x - cos 2x) dx?

Question

Integrate from pi/4 to pi for sec x (sec x - cos 2x) dx?

Answer 1

int(pi / 4, pi) [ sec x (sec x - cos 2x) ] dx

= int(pi / 4, pi) [ sec^2(x) - sec(x) (2 cos^2(x) - 1) ] dx

= int(pi / 4, pi) [ sec^2(x) - 2 cos(x) + sec(x) ] dx

= [ tan(x) - 2 sin(x) + ln | sec(x) + tan(x) | ] (pi / 4, pi)

=  tan(pi) - 2 sin(pi) + ln | sec(pi) + tan(pi) | - tan(pi / 4) + 2 sin(pi / 4) - ln | sec(pi / 4) + tan(pi / 4) |

= - 1 + 2 / sqrt(2) - ln(sqrt(2) + 1)

= sqrt(2) - 1 - ln( sqrt(2) + 1 ).

The integration of sec(x) is explained here:

http://www.mecca.org/~halfacre/MATH/less...