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

1. Proof

    Strategy: Use definition of sinh.
     
    sinh x =
    ex - e-x
    2
    (integral) sinh x dx = (integral)
    ex - ex
    2
    dx
    = (1/2) (integral) ex dx - (1/2) (integral) ex dx
    solve left equation
    = (1/2) ex - (1/2) (integral) ex dx

    set
      u = - x
    then we find
      du = - dx

    substitute du= - dx, u= - x

    = (1/2) ex + (1/2) (integral) - ex dx
    = (1/2) ex + (1/2) (integral) eu du

    solve the right integral
    = (1/2) ex + (1/2) eu + C

    substitute back u= - x
    = (1/2) ex + (1/2) ex + C
    ex + ex
    2
    + C
    which by definition
    = cosh x + C
    Q.E.D.

  
 
  

 
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