learn how to integrate tan x

integral of tan x
Strategy: Make in terms of sin's and cos's; Use Substitution.
 

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

Alternatively,

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


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