What is the integral of 1/(log_{e)(x)} ?

The integral of 1/(log_{e)(x)} is \Ei(ln(x))+C

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integral of 1/(log_{e)(x)}

\int \frac{1}{lo g _{e} ( x )} d x

Ei (ln (x)) + C

Solution steps

\int \frac{1}{lo g _{e} ( x )} d x

Simplify

lo g_{e} (x) : ln (x)

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

Apply u-substitution

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

Use the nonelementary integral:

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

= Ei (u)

Substitute back

u = ln (x)

= Ei (ln (x))

Add a constant to the solution

= Ei (ln (x)) + C

What is the integral of 1/(log_{e)(x)} ?
The integral of 1/(log_{e)(x)} is \Ei(ln(x))+C