Integral Sinh(x)

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Math2.org Math Tables: Integral sinh(x)
(Math)
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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