Dave's Math Tables: Integral tan(x)

(Math | Calculus | Integrals | Table Of | tan x)

Discussion of tan x = - ln|cos x| + C.

1. Proof

Strategy: Make in terms of sin's and cos's; Use Subtitution.
 

tan x dx =

sin x cos x

dx

set
  u = cos x.
then we find
  du = - sin x dx

substitute du=-sin x, u=cos x
 

sin x cos x

dx = -

(-1) sin x dx cos x

 

= -

du u

Solve the integral

= - ln |u| + C

substitute back u=cos x

= - ln |cos x| + C
Q.E.D.

2. Alternate Form of Result

tan x dx = - ln |cos x| + C
= ln | (cos x)^-1 | + C
= ln |sec x| + C

Therefore:

tan x dx = - ln |cos x| + C = ln |sec x| + C