How to calculate the integral of sec(x)?

First of all, multiply secx by (secx+tanx)/(secx+tanx). Use the substitution u=secx+tanx, so that du=(secxtanx+sec²x) dx and then substitute both terms. Calculate the integral of the du/u arriving at ln|u|+C. Then put in the substituted function of x. The result is ln|secx+tanx|+C.