What is the integral from 1/2 to 2 of |ln(x)| ?

The integral from 1/2 to 2 of |ln(x)| is -1/2+3/2 ln(2)

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integral from 1/2 to 2 of |ln(x)|

\int_{\frac{1}{2}}^{2} ∣ ln (x) ∣ d x

- \frac{1}{2} + \frac{3}{2} ln (2)

Decimal Notation

0.53972 \dots

Solution steps

\int_{\frac{1}{2}}^{2} ∣ ln (x) ∣ d x

Apply u-substitution

: \int_{- l n (2)}^{l n (2)} ∣ u ∣ e^{u} d u

= \int_{- l n (2)}^{l n (2)} ∣ u ∣ e^{u} d u

Eliminate Absolutes

= \int_{- l n (2)}^{0} (- u) e^{u} d u + \int_{0}^{l n (2)} u e^{u} d u

\int_{- l n (2)}^{0} (- u) e^{u} d u = \frac{1}{2} - \frac{1}{2} ln (2)

\int_{0}^{l n (2)} u e^{u} d u = 2 ln (2) - 1

= \frac{1}{2} - \frac{1}{2} ln (2) + 2 ln (2) - 1

Simplify

\frac{1}{2} - \frac{1}{2} ln (2) + 2 ln (2) - 1 : - \frac{1}{2} + \frac{3}{2} ln (2)

= - \frac{1}{2} + \frac{3}{2} ln (2)

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What is the integral from 1/2 to 2 of |ln(x)| ?
The integral from 1/2 to 2 of |ln(x)| is -1/2+3/2 ln(2)