What is the integral of 2tanh(x) ?

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

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integral of 2tanh(x)

\int 2 tanh (x) d x

2 ln ∣ cosh (x) ∣ + C

Solution steps

\int 2 tanh (x) d x

Take the constant out:

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

= 2 \cdot \int tanh (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = cosh (x)

= 2 ln ∣ cosh (x) ∣

Add a constant to the solution

= 2 ln ∣ cosh (x) ∣ + C

ln (7)

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