Integral coth(x)

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Proof: Integral coth(x)

(Math | Calculus | Integrals | Table Of | coth x)

Discussion of (integral)

coth x dx = ln |sinh x| + C.

1. Proof

Strategy

coth x =

cosh x

sinh x

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

coth x dx = (integral)

e^x + e^-xe^x - e^-x

^-x

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

= ln |2 sinh x| + C
= ln 2 + ln |sinh x| + C

ln 2 is merely a constant that can be combined with C

= ln |sinh x| + C
Q.E.D.