What is the integral of 1/(tanh(3x)) ?

The integral of 1/(tanh(3x)) is 1/3 ln|sinh(3x)|+C

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integral of 1/(tanh(3x))

\int \frac{1}{tanh ( 3 x )} d x

\frac{1}{3} ln ∣ sinh (3 x) ∣ + C

Solution steps

\int \frac{1}{tanh ( 3 x )} d x

Use the Hyperbolic identity:

\frac{1}{tanh ( x )} = coth (x)

= \int coth (3 x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int \frac{1}{3 u} d u

= \int \frac{1}{3 u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = sinh (3 x)

= \frac{1}{3} ln ∣ sinh (3 x) ∣

Add a constant to the solution

= \frac{1}{3} ln ∣ sinh (3 x) ∣ + C

3 y^{^{''}} + 75 y = 81 sin (4 t)

y = x^{3}, (2, 8)

y^{^{''}} + 8 y^{^{'}} + 16 y = 2 e^{- 4 t}

\frac{\partial}{\partial x} (3 x ln (x y))

\frac{\partial}{\partial x} ((x^{3} + x^{4}) (x^{2} + 3 x)^{2})

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