What is the integral of coth(x/3) ?

The integral of coth(x/3) is 3ln|sinh(x/3)|+C

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integral of coth(x/3)

\int coth (\frac{x}{3}) d x

3 ln sinh (\frac{x}{3}) + C

Solution steps

\int coth (\frac{x}{3}) d x

Apply u-substitution

: \int coth (u) \cdot 3 d u

= \int coth (u) \cdot 3 d u

Take the constant out:

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

= 3 \cdot \int coth (u) d u

Rewrite using trig identities

= 3 \cdot \int \frac{cosh ( u )}{sinh ( u )} d u

Apply u-substitution

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

= 3 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 ln ∣ v ∣

Substitute back

= 3 ln sinh (\frac{x}{3})

Add a constant to the solution

= 3 ln sinh (\frac{x}{3}) + C

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What is the integral of coth(x/3) ?
The integral of coth(x/3) is 3ln|sinh(x/3)|+C