How do I figure out the integral of sec(x) dx? My mind is blanking. :oops:
P pamw New member Joined Jan 10, 2007 Messages 8 Feb 23, 2007 #1 How do I figure out the integral of sec(x) dx? My mind is blanking.
G galactus Super Moderator Staff member Joined Sep 28, 2005 Messages 7,216 Feb 23, 2007 #2 One way is a little trick: \(\displaystyle \L\\\int{sec(x)}dx=\int{sec(x)\frac{sec(x)+tan(x)}{sec(x)+tan(x)}}dx\) =\(\displaystyle \L\\\int\frac{sec^{2}(x)+sec(x)tan(x)}{sec(x)+tan(x)}dx\) Let \(\displaystyle u=sec(x)+tan(x), \;\ du=(sec^{2}(x)+sec(x)tan(x))dx\) \(\displaystyle \L\\\int\frac{1}{u}du=ln|u|\) \(\displaystyle \L\\=ln|sec(x)+tan(x)|+C\)
One way is a little trick: \(\displaystyle \L\\\int{sec(x)}dx=\int{sec(x)\frac{sec(x)+tan(x)}{sec(x)+tan(x)}}dx\) =\(\displaystyle \L\\\int\frac{sec^{2}(x)+sec(x)tan(x)}{sec(x)+tan(x)}dx\) Let \(\displaystyle u=sec(x)+tan(x), \;\ du=(sec^{2}(x)+sec(x)tan(x))dx\) \(\displaystyle \L\\\int\frac{1}{u}du=ln|u|\) \(\displaystyle \L\\=ln|sec(x)+tan(x)|+C\)
