What is the integral of tanh((2x)/3) ?

The integral of tanh((2x)/3) is 3/2 ln|cosh((2x)/3)|+C

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integral of tanh((2x)/3)

\int tanh (\frac{2 x}{3}) d x

\frac{3}{2} ln cosh (\frac{2 x}{3}) + C

Solution steps

\int tanh (\frac{2 x}{3}) d x

Apply u-substitution

: \int tanh (u) \frac{3}{2} d u

= \int tanh (u) \frac{3}{2} d u

Take the constant out:

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

= \frac{3}{2} \cdot \int tanh (u) d u

Rewrite using trig identities

= \frac{3}{2} \cdot \int \frac{sinh ( u )}{cosh ( u )} d u

Apply u-substitution

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

= \frac{3}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{3}{2} ln ∣ v ∣

Substitute back

= \frac{3}{2} ln cosh (\frac{2 x}{3})

Add a constant to the solution

= \frac{3}{2} ln cosh (\frac{2 x}{3}) + C

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What is the integral of tanh((2x)/3) ?
The integral of tanh((2x)/3) is 3/2 ln|cosh((2x)/3)|+C