Integral tanh(x)

Themes > Science > Mathematics > Calculus > Integrals > Integrals Table > Integral tanh(x)

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

(e^x - e^-x) / 2

(e^x + e^-x) / 2

tanh x dx = (integral)

e^x - e^-x

e^x + e^-x

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

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

= (integral) ^{du

u}

solve

= ln |u| + C

substitute back u = e^x + e^-x

= ln |e^x + e^-x| + C

since e^x and e^-x are always positive

= ln (e^x + e^-x) + C

since (e^x + 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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