What is the integral from 0 to 0.5 of 1/(xln^2|x|) ?

The integral from 0 to 0.5 of 1/(xln^2|x|) is 1.44269…

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integral from 0 to 0.5 of 1/(xln^2|x|)

\int_{0}^{0.5} \frac{1}{x ln ^{2} ∣ x ∣} d x

1.44269 \dots

Solution steps

\int_{0}^{0.5} \frac{1}{x ln ∣ x ∣ ^{2}} d x

Eliminate Absolutes

= \int_{0}^{0.5} \frac{1}{x ln ^{2} ( x )} d x

Apply Integration By Parts

= \frac{1 - ln ^{2} ( x ) \cdot \int - \frac{l n ( x ) + 2}{x l n ^{3} ( x )} d x}{ln ^{2} ( x )}_{0}^{0.5}

\int - \frac{ln ( x ) + 2}{x ln ^{3} ( x )} d x = \frac{1}{ln ( x )} + \frac{1}{ln ^{2} ( x )}

= \frac{1 - ln ^{2} ( x ) ( \frac{1}{l n ( x )} + \frac{1}{l n ^{2} ( x )} )}{ln ^{2} ( x )}_{0}^{0.5}

\frac{1 - ln ^{2} ( x ) ( \frac{1}{l n ( x )} + \frac{1}{l n ^{2} ( x )} )}{ln ^{2} ( x )} = - \frac{1}{ln ( x )}

= [- \frac{1}{ln ( x )}]_{0}^{0.5}

Compute the boundaries:

1.44269 \dots

= 1.44269 \dots

What is the integral from 0 to 0.5 of 1/(xln^2|x|) ?
The integral from 0 to 0.5 of 1/(xln^2|x|) is 1.44269…