Integrals

The integrals of the trig functions are:
$\int \sin = -\cos$
$\int \cos = \sin$
$\int \tan = \log \abs{\sec}$
  ($= -\log \abs{\cos}$)
$\int \sec = \log \abs{\sec + \tan}$

...and the co-s:
$\int \cot = \log \abs{\sin}$
  ($= -\log \abs{\csc}$)
$\int \csc = -\log \abs{\csc + \cot}$

$\sin$ and $\cos$ follow from inverting the derivative;
$\tan$ follows from the $u$-substitution $u=\cos$;
  this is only valid on $(-\pi/2,\pi/2)$, and the absolute values are
  to compensate for this (on the odd intervals, the $u$-sub is
  negative, hence you need a compensating sign; alternately, you just
  need to make $\sec$ always positive, as $\tan$ has period $\pi$, not
  $2\pi$).
$\sec$ is harder. One can check it, obviously.

Indeed, $\int \sec$ has a special name: it's the inverse Gudermannian,
which shows up in the Mercator projection,
and which also gives a way to switch between trig functions and
hyperbolic functions w/o using imaginary numbers
(this is sorta weird):
$\sin \gd x = \tanh x$, for instance.

(This was useful for tables of values, for instance.)

The modern notation for hyperbolic functions is due to Lampert, who
also popularized them.
($Sc,Cc$ and $Sh,Ch$ were old notations.)

The Gudermannian is also called
- "the transcendental angle"
- "the hyperbolic amplitude" 
