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

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

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integral of tanh(1-3x)

\int tanh (1 - 3 x) d x

- \frac{1}{3} ln ∣ cosh (1 - 3 x) ∣ + C

Solution steps

\int tanh (1 - 3 x) d x

Apply u-substitution

: \int - \frac{1}{3} tanh (u) d u

= \int - \frac{1}{3} tanh (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 tanh (u) d u

Rewrite using trig identities

= - \frac{1}{3} \cdot \int \frac{sinh ( u )}{cosh ( u )} d u

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - \frac{1}{3} ln ∣ v ∣

Substitute back

= - \frac{1}{3} ln ∣ cosh (1 - 3 x) ∣

Add a constant to the solution

= - \frac{1}{3} ln ∣ cosh (1 - 3 x) ∣ + C

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