What is the integral of tanh(x/7) ?

The integral of tanh(x/7) is 7ln|cosh(x/7)|+C

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integral of tanh(x/7)

\int tanh (\frac{x}{7}) d x

7 ln cosh (\frac{x}{7}) + C

Solution steps

\int tanh (\frac{x}{7}) d x

Apply u-substitution

: \int tanh (u) \cdot 7 d u

= \int tanh (u) \cdot 7 d u

Take the constant out:

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

= 7 \cdot \int tanh (u) d u

Rewrite using trig identities

= 7 \cdot \int \frac{sinh ( u )}{cosh ( u )} d u

Apply u-substitution

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

= 7 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 7 ln ∣ v ∣

Substitute back

= 7 ln cosh (\frac{x}{7})

Add a constant to the solution

= 7 ln cosh (\frac{x}{7}) + C

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What is the integral of tanh(x/7) ?
The integral of tanh(x/7) is 7ln|cosh(x/7)|+C