What is the integral of tanh(x/9) ?

The integral of tanh(x/9) is 9ln|cosh(x/9)|+C

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integral of tanh(x/9)

\int tanh (\frac{x}{9}) d x

9 ln cosh (\frac{x}{9}) + C

Solution steps

\int tanh (\frac{x}{9}) d x

Apply u-substitution

= \int tanh (u) \cdot 9 d u

Take the constant out:

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

= 9 \cdot \int tanh (u) d u

Rewrite using trig identities

= 9 \cdot \int \frac{sinh ( u )}{cosh ( u )} d u

Apply u-substitution

= 9 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 9 ln ∣ v ∣

Substitute back

= 9 ln cosh (\frac{x}{9})

Add a constant to the solution

= 9 ln cosh (\frac{x}{9}) + C

What is the integral of tanh(x/9) ?
The integral of tanh(x/9) is 9ln|cosh(x/9)|+C