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Proof: Integral tan(x)
(Math | Calculus | Integrals | Table Of | tan x)
 
Discussion of
(integral) tan x = - ln|cos x| + C.

1. Proof

    Strategy: Make in terms of sin's and cos's; Use Substitution.
     
    (integral) tan x dx = (integral) sin x
    COs x
    dx
    set
      u = COs x.
    then we find
      du = - sin x dx

    substitute du=-sin x, u=COs x
     
    (integral) sin x
    COs x
    dx = - (integral)
    (-1) sin x dx
    COs x
     
    = - (integral) 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

    (integral) tan x dx = - ln |COs x| + C
    = ln | (COs x)-1 | + C
    = ln |sec x| + C

    Therefore:
    (integral) tan x dx = - ln |COs x| + C = ln |sec x| + C

  
 
  

 
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