What is the integral of (tanh(ln(x)))/x ?

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

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integral of (tanh(ln(x)))/x

\int \frac{tanh ( ln ( x ) )}{x} d x

ln ∣ cosh (ln (x)) ∣ + C

Solution steps

\int \frac{tanh ( ln ( x ) )}{x} d x

Apply u-substitution

: \int tanh (u) d u

= \int tanh (u) d u

Rewrite using trig identities

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= ln ∣ v ∣

Substitute back

= ln ∣ cosh (ln (x)) ∣

Add a constant to the solution

= ln ∣ cosh (ln (x)) ∣ + C

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