What is the integral from 0 to 1 of x/(ln(x)) ?

The integral from 0 to 1 of x/(ln(x)) is \Ei(2ln(1))

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integral from 0 to 1 of x/(ln(x))

\int_{0}^{1} \frac{x}{ln ( x )} d x

Ei (2 ln (1))

Solution steps

\int_{0}^{1} \frac{x}{ln ( x )} d x

Apply Integration By Parts

: [\frac{x ^{2}}{2 ln ( x )} - \int - \frac{x}{2 ln ^{2} ( x )} d x]_{0}^{1}

= [\frac{x ^{2}}{2 ln ( x )} - \int - \frac{x}{2 ln ^{2} ( x )} d x]_{0}^{1}

\int - \frac{x}{2 ln ^{2} ( x )} d x = \frac{x ^{2}}{2 ln ( x )} - Ei (2 ln (x))

= [\frac{x ^{2}}{2 ln ( x )} - (\frac{x ^{2}}{2 ln ( x )} - Ei (2 ln (x)))]_{0}^{1}

Simplify

= [Ei (2 ln (x))]_{0}^{1}

Compute the boundaries:

Ei (2 ln (1))

= Ei (2 ln (1))

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What is the integral from 0 to 1 of x/(ln(x)) ?
The integral from 0 to 1 of x/(ln(x)) is \Ei(2ln(1))