What is the integral of tanh(ax) ?

The integral of tanh(ax) is 1/a ln|cosh(ax)|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of tanh(ax)

\int tanh (a x) d x

\frac{1}{a} ln ∣ cosh (a x) ∣ + C

Solution steps

\int tanh (a x) d x

Rewrite using trig identities

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

Apply u-substitution

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{a} ln ∣ u ∣

Substitute back

u = cosh (a x)

= \frac{1}{a} ln ∣ cosh (a x) ∣

Add a constant to the solution

= \frac{1}{a} ln ∣ cosh (a x) ∣ + C

What is the integral of tanh(ax) ?
The integral of tanh(ax) is 1/a ln|cosh(ax)|+C