首先求∫sec^3(x) dx：記I=∫sec^3(x) dx，則I

=∫sec(x)*sec^2(x) dx

=∫sec(x)*[tan(x)]" dx

=sec(x)*tan(x)-∫[sec(x)]"*tan(x) dx

=sec(x)*tan(x)-∫[sec(x)*tan(x)]*tan(x) dx

=sec(x)*tan(x)-∫sec(x)*tan^2(x) dx

=sec(x)*tan(x)-∫sec(x)*[sec^2(x)-1] dx

=sec(x)*tan(x)-∫sec^3(x) dx+∫sec(x) dx

=sec(x)*tan(x)-I+ln|sec(x)+tan(x)|+C，

所以2I=sec(x)*tan(x)+ln|sec(x)+tan(x)|+C，

I=sec(x)*tan(x)/2+ln|sec(x)+tan(x)|/2+C，C為任意常數

然後求∫sec^5(x) dx：記J=∫sec^5(x) dx，則J

=∫sec^3(x)*sec^2(x) dx

=∫sec^3(x)*[tan(x)]" dx

=sec^3(x)*tan(x)-∫[sec^3(x)]"*tan(x) dx

=sec^3(x)*tan(x)-∫3sec^2(x)*[sec(x)*tan(x)]*tan(x) dx

=sec^3(x)*tan(x)-3∫sec^3(x)*tan^2(x) dx

=sec^3(x)*tan(x)-3∫sec^3(x)*[sec^2(x)-1] dx

=sec^3(x)*tan(x)-3∫sec^5(x) dx+3∫sec^3(x) dx

=sec^3(x)*tan(x)-3J+3I，

所以4J=sec^3(x)*tan(x)+3I，

J=sec^3(x)*tan(x)/4+3I/4

=sec^3(x)*tan(x)/4+3sec(x)*tan(x)/8+3ln|sec(x)+tan(x)|/8+C，

C為任意常數