What is the integral of 1/(ln(2x)) ?

The integral of 1/(ln(2x)) is 1/2 \Ei(ln(2x))+C

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integral of 1/(ln(2x))

\int \frac{1}{ln ( 2 x )} d x

\frac{1}{2} Ei (ln (2 x)) + C

Solution steps

\int \frac{1}{ln ( 2 x )} d x

Apply u-substitution

= \int \frac{1}{2 ln ( u )} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{ln ( u )} d u

Apply u-substitution

= \frac{1}{2} \cdot \int \frac{e ^{v}}{v} d v

Use the nonelementary integral:

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

= \frac{1}{2} Ei (v)

Substitute back

= \frac{1}{2} Ei (ln (2 x))

Add a constant to the solution

= \frac{1}{2} Ei (ln (2 x)) + C

What is the integral of 1/(ln(2x)) ?
The integral of 1/(ln(2x)) is 1/2 \Ei(ln(2x))+C