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

1. Proof

    Strategy: Use definition of tanh; Use Substitution.
    tanh x =
    sinh x 
    cosh x
    =
    (ex - e-x) / 2 
    (ex + ex) / 2
     
    (integral) tanh x dx = (integral)
    ex - ex 
    ex + ex
    dx

    set
      u = ex + ex
    then we find
      du = (ex - ex) dx

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

    solve

    = ln |u| + C

    substitute back u = ex + ex

    = ln |ex + ex| + C

    since ex and ex are always positive

    = ln (ex + ex) + C

    since (ex + ex)/2 = cosh(x)

    = ln (2 cosh x) + C
    = ln 2 + ln (cosh x) + C

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

    = ln (cosh x) + C
    Q.E.D.

  
 
  

 
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