What is the integral of tanh(x) ?

The integral of tanh(x) is ln|cosh(x)|+C

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integral of tanh(x)

\int tanh (x) d x

ln ∣ cosh (x) ∣ + C

Solution steps

\int tanh (x) d x

Rewrite using trig identities

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

= ln ∣ cosh (x) ∣

Add a constant to the solution

= ln ∣ cosh (x) ∣ + C

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