Proof: Integral Tanh(x)

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Strategy: Use definition of tanh; Use Substitution.
tanh x = sinh x cosh x = (ex - e-x) / 2 (ex + ex) / 2
(integral) tanh x dx = (integral) ex - ex ex + ex dx

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

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

= (integral) du u

solve

= ln |u| + C

substitute back u = ex + ex

= ln |ex + ex| + C

since ex and ex are always positive

= ln (ex + ex) + C since (ex + ex)/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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