What is the integral of csch(6x) ?

The integral of csch(6x) is 1/6 ln(tanh(3x))+C

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integral of csch(6x)

\int csch (6 x) d x

\frac{1}{6} ln (tanh (3 x)) + C

Solution steps

\int csch (6 x) d x

Apply u-substitution

: \int csch (u) \frac{1}{6} d u

= \int csch (u) \frac{1}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int csch (u) d u

Use the common integral:

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

= \frac{1}{6} ln (tanh (\frac{u}{2}))

Substitute back

u = 6 x

= \frac{1}{6} ln (tanh (\frac{6 x}{2}))

Simplify

\frac{1}{6} ln (tanh (\frac{6 x}{2})) : \frac{1}{6} ln (tanh (3 x))

= \frac{1}{6} ln (tanh (3 x))

Add a constant to the solution

= \frac{1}{6} ln (tanh (3 x)) + C

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What is the integral of csch(6x) ?
The integral of csch(6x) is 1/6 ln(tanh(3x))+C