∫sec³x dx

By Integration by Parts:

U = secx ................dv = sec²x dx

du = secxtanxdx.......v = tanx


= secxtanx- ∫secxtan²x dx // [tan²x = (sec²x-1)]

= secxtanx- ∫ secx(sec²x -1)dx

= secxtanx- ∫ sec³x dx + ∫secx dx

= secxtanx + ln|secxtanx| - ∫sec³x dx = ∫sec³x dx

= secxtanx + ln|secxtanx|= 2∫sec³x dx


= (1/2)secxtanx + (1/2)ln|sexctanx| = ∫sec³x dx


∫sec³x dx = (1/2)secxtanx + (1/2)ln|sexctanx| + C