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

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

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integral of x/(ln(x))

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

Ei (2 ln (x)) + C

Solution steps

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

Apply u-substitution

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

Apply u-substitution

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

Use the nonelementary integral:

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

= Ei (v)

Substitute back

= Ei (2 ln (x))

Add a constant to the solution

= Ei (2 ln (x)) + C