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

The integral from 0 to e of (ln(x))/x is diverges

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

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

diverges

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Compute the boundaries:

diverges

= diverges

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What is the integral from 0 to e of (ln(x))/x ?
The integral from 0 to e of (ln(x))/x is diverges