What is the integral of 4tanh(3x) ?

The integral of 4tanh(3x) is 4/3 ln|cosh(3x)|+C

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integral of 4tanh(3x)

\int 4 tanh (3 x) d x

\frac{4}{3} ln ∣ cosh (3 x) ∣ + C

Solution steps

\int 4 tanh (3 x) d x

Take the constant out:

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

= 4 \cdot \int tanh (3 x) d x

Rewrite using trig identities

= 4 \cdot \int \frac{sinh ( 3 x )}{cosh ( 3 x )} d x

Apply u-substitution

= 4 \cdot \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

= 4 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = cosh (3 x)

= 4 \cdot \frac{1}{3} ln ∣ cosh (3 x) ∣

Simplify

4 \cdot \frac{1}{3} ln ∣ cosh (3 x) ∣ : \frac{4}{3} ln ∣ cosh (3 x) ∣

= \frac{4}{3} ln ∣ cosh (3 x) ∣

Add a constant to the solution

= \frac{4}{3} ln ∣ cosh (3 x) ∣ + C

What is the integral of 4tanh(3x) ?
The integral of 4tanh(3x) is 4/3 ln|cosh(3x)|+C