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Proof: Integral sec(x)
(Math | Calculus | Integrals | Table Of | 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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