What is the integral of csch(2z) ?

The integral of csch(2z) is 1/2 ln(tanh(z))+C

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integral of csch(2z)

\int csch (2 z) d z

\frac{1}{2} ln (tanh (z)) + C

Solution steps

\int csch (2 z) d z

Apply u-substitution

: \int csch (u) \frac{1}{2} d u

= \int csch (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int csch (u) d u

Use the common integral:

\int csch (u) d u = ln (tanh (\frac{u}{2}))

= \frac{1}{2} ln (tanh (\frac{u}{2}))

Substitute back

u = 2 z

= \frac{1}{2} ln (tanh (\frac{2 z}{2}))

Simplify

\frac{1}{2} ln (tanh (\frac{2 z}{2})) : \frac{1}{2} ln (tanh (z))

= \frac{1}{2} ln (tanh (z))

Add a constant to the solution

= \frac{1}{2} ln (tanh (z)) + C

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What is the integral of csch(2z) ?
The integral of csch(2z) is 1/2 ln(tanh(z))+C