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

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

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integral of 2/(ln(x))

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

2 Ei (ln (x)) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \int \frac{e ^{u}}{u} d u

Use the nonelementary integral:

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

= 2 Ei (u)

Substitute back

u = ln (x)

= 2 Ei (ln (x))

Add a constant to the solution

= 2 Ei (ln (x)) + C

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