What is the integral of tanh(2x) ?

The integral of tanh(2x) is 1/2 ln|cosh(2x)|+C

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integral of tanh(2x)

\int tanh (2 x) d x

\frac{1}{2} ln ∣ cosh (2 x) ∣ + C

Solution steps

\int tanh (2 x) d x

Apply u-substitution

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

= \int tanh (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 tanh (u) d u

Rewrite using trig identities

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

= \frac{1}{2} ln ∣ cosh (2 x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ cosh (2 x) ∣ + C

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\frac{d}{d x} (2^{l n (x)})

What is the integral of tanh(2x) ?
The integral of tanh(2x) is 1/2 ln|cosh(2x)|+C