Integral Tanh(x)

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Math2.org Math Tables: Integral tanh(x)
(Math)
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 + e-x) / 2
    (integral) tanh x dx = (integral) ex - e-x ex + e-x dx

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

    substitute du= (ex - e-x) dx, u = ex + e-x

    = (integral) du u

    solve

    = ln |u| + C

    substitute back u = ex + e-x

    = ln |ex + e-x| + C

    since ex and e-x are always positive

    = ln (ex + e-x) + C since (ex + e-x)/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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