Themes > Science > Mathematics > Calculus > Integrals > Integrals Table > Integral sec(x)

Discussion of (integral) sec x = ln|sec x + tan x| + C.


1. Proof

    Strategy: The strategy is not obvious.  Multiply and divide by (sec x + tan x); use Substitution.
     
    (integral) sec x dx = (integral) sec x  sec x + tan x 

    sec x + tan x

    		 dx
    set
      u = sec x + tan x
    then we find
      du = (sec x tan x + sec2 x) dx

    substitute du = (sec x tan x + sec2 x) dx, u = sec x + tan x
     
    (integral) sec x  sec x + tan x 

    sec x + tan x

    		 dx = (integral)
    (sec2 x + sec x tan x) dx 

    sec x + tan x

     
    = (integral) 
    du 

    u

    solve integral

    = ln |u| + C

    substitute back u=sec x + tan x

    = ln |sec x + tan x| + C
    Q.E.D.


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