Integral Sinh(x)

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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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