What is the integral of 1/(ln(3x)) ?

The integral of 1/(ln(3x)) is 1/3 \Ei(ln(3x))+C

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integral of 1/(ln(3x))

\int \frac{1}{ln ( 3 x )} d x

\frac{1}{3} Ei (ln (3 x)) + C

Solution steps

\int \frac{1}{ln ( 3 x )} d x

Apply u-substitution

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

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

Apply u-substitution

: \int \frac{e ^{v}}{v} d v

= \frac{1}{3} \cdot \int \frac{e ^{v}}{v} d v

Use the nonelementary integral:

\int \frac{e ^{x}}{x} d x = Ei (x)

= \frac{1}{3} Ei (v)

Substitute back

= \frac{1}{3} Ei (ln (3 x))

Add a constant to the solution

= \frac{1}{3} Ei (ln (3 x)) + C

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What is the integral of 1/(ln(3x)) ?
The integral of 1/(ln(3x)) is 1/3 \Ei(ln(3x))+C