What is the integral from 0 to 1/2 of 1/(xln(x)) ?

The integral from 0 to 1/2 of 1/(xln(x)) is diverges

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integral from 0 to 1/2 of 1/(xln(x))

\int_{0}^{\frac{1}{2}} \frac{1}{x ln ( x )} d x

diverges

Solution steps

\int_{0}^{\frac{1}{2}} \frac{1}{x ln ( x )} d x

Apply Integration By Parts

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

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

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

= [\frac{1 - ln ( x ) ( - ln ∣ ln ( x ) ∣ + \frac{1}{l n ( x )} )}{ln ( x )}]_{0}^{\frac{1}{2}}

\frac{1 - ln ( x ) ( - ln ∣ ln ( x ) ∣ + \frac{1}{l n ( x )} )}{ln ( x )} = ln ∣ ln (x) ∣

= [ln ∣ ln (x) ∣]_{0}^{\frac{1}{2}}

Compute the boundaries:

diverges

= diverges

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What is the integral from 0 to 1/2 of 1/(xln(x)) ?
The integral from 0 to 1/2 of 1/(xln(x)) is diverges