What is the integral of coth(u/(sqrt(2))) ?

The integral of coth(u/(sqrt(2))) is sqrt(2)ln|sinh(u/(sqrt(2)))|+C

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integral of coth(u/(sqrt(2)))

\int coth (\frac{u}{2}) d u

2 ln sinh (\frac{u}{2}) + C

Solution steps

\int coth (\frac{u}{2}) d u

Apply u-substitution

: \int coth (v) 2 d v

= \int coth (v) 2 d v

Take the constant out:

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

= 2 \cdot \int coth (v) d v

Rewrite using trig identities

= 2 \cdot \int \frac{cosh ( v )}{sinh ( v )} d v

Apply u-substitution

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

= 2 \cdot \int \frac{1}{w} d w

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= 2 ln ∣ w ∣

Substitute back

= 2 ln sinh (\frac{u}{2})

Add a constant to the solution

= 2 ln sinh (\frac{u}{2}) + C

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What is the integral of coth(u/(sqrt(2))) ?
The integral of coth(u/(sqrt(2))) is sqrt(2)ln|sinh(u/(sqrt(2)))|+C