What is the integral of coth(y) ?

The integral of coth(y) is ln|sinh(y)|+C

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integral of coth(y)

\int coth (y) d y

ln ∣ sinh (y) ∣ + C

Solution steps

\int coth (y) d y

Rewrite using trig identities

= \int \frac{cosh ( y )}{sinh ( y )} d y

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = sinh (y)

= ln ∣ sinh (y) ∣

Add a constant to the solution

= ln ∣ sinh (y) ∣ + C

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