What is the integral of coth(5x) ?

The integral of coth(5x) is 1/5 ln|sinh(5x)|+C

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integral of coth(5x)

\int coth (5 x) d x

\frac{1}{5} ln ∣ sinh (5 x) ∣ + C

Solution steps

\int coth (5 x) d x

Rewrite using trig identities

= \int \frac{cosh ( 5 x )}{sinh ( 5 x )} d x

Apply u-substitution

= \int \frac{1}{5 u} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = sinh (5 x)

= \frac{1}{5} ln ∣ sinh (5 x) ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ sinh (5 x) ∣ + C

What is the integral of coth(5x) ?
The integral of coth(5x) is 1/5 ln|sinh(5x)|+C