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Proof: Integral coth(x)
(Math | Calculus | Integrals | Table Of | coth x)
 
Discussion of
(integral) coth x dx = ln |sinh x| + C.

1. Proof

    Strategy: Use definition of coth; Use Substitution.
    coth x =
    cosh x 
    sinh x
    =
    (ex + e-x) / 2 
    (ex - e-x) / 2
     
    (integral) coth x dx = (integral)
    ex + e-x 
    ex - e-x
    dx
    set
      u = ex - e-x
    then we find
      du = (ex + e-x) dx

    substitute du= (ex + e-x) dx, u = ex - e-x
     
    (integral)
    du 
    u
    solve

    = ln |u| + C

    substitute back u = ex - e-x

    = ln |ex - e-x| + C

    since (ex - e-x)/2 = sinh(x)

    = ln |2 sinh x| + C
    = ln 2 + ln |sinh x| + C

    ln 2 is merely a constant that can be combined with C

    = ln |sinh x| + C
    Q.E.D.

  
 
  

 
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