Themes > Science > Mathematics > Calculus > Integrals > Integrals Table > Integral 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 - e-x

    2

    		 dx
    = (1/2) (integral) ex dx - (1/2) (integral) e-x dx
    solve left equation
    = (1/2) ex - (1/2) (integral) e-x dx

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

    substitute du= - dx, u= - x

    = (1/2) ex + (1/2) (integral) - e-x 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) e-x + C
    = 
    ex + e-x

    2

    		 + C
    which by definition
    = cosh x + C
    Q.E.D.


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