What is the integral from 0.5 to 1 of ln|x-0.5| ?

The integral from 0.5 to 1 of ln|x-0.5| is -0.84657…

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integral from 0.5 to 1 of ln|x-0.5|

\int_{0.5}^{1} ln ∣ x - 0.5 ∣ d x

- 0.84657 \dots

Solution steps

\int_{0.5}^{1} ln ∣ x - 0.5 ∣ d x

Eliminate Absolutes

= \int_{0.5}^{1} ln (x - 0.5) d x

Apply u-substitution

: \int_{0}^{0.5} ln (u) d u

= \int_{0}^{0.5} ln (u) d u

Apply Integration By Parts

: [u ln (u) - \int 1 d u]_{0}^{0.5}

= [u ln (u) - \int 1 d u]_{0}^{0.5}

\int 1 d u = u

= [u ln (u) - u]_{0}^{0.5}

Compute the boundaries:

- 0.84657 \dots

= - 0.84657 \dots

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\int_{- 1}^{1} x x^{2} + 3 d x

\int_{- 1}^{1} 3 x^{2} + 4 d x

What is the integral from 0.5 to 1 of ln|x-0.5| ?
The integral from 0.5 to 1 of ln|x-0.5| is -0.84657…