What is the integral of 1/(tanh(x)) ?

The integral of 1/(tanh(x)) is ln|sinh(x)|+C

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integral of 1/(tanh(x))

\int \frac{1}{tanh ( x )} d x

ln ∣ sinh (x) ∣ + C

Solution steps

\int \frac{1}{tanh ( x )} d x

Use the Hyperbolic identity:

\frac{1}{tanh ( x )} = coth (x)

= \int coth (x) d x

Rewrite using trig identities

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

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 (x)

= ln ∣ sinh (x) ∣

Add a constant to the solution

= ln ∣ sinh (x) ∣ + C

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What is the integral of 1/(tanh(x)) ?
The integral of 1/(tanh(x)) is ln|sinh(x)|+C