What is the integral from 0 to 1 of ln(sqrt(x)) ?

The integral from 0 to 1 of ln(sqrt(x)) is -1/2

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integral from 0 to 1 of ln(sqrt(x))

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

- \frac{1}{2}

Decimal Notation

- 0.5

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{1}{2} d x = \frac{1}{2} x

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

Simplify

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

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

Compute the boundaries:

- \frac{1}{2}

= - \frac{1}{2}

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What is the integral from 0 to 1 of ln(sqrt(x)) ?
The integral from 0 to 1 of ln(sqrt(x)) is -1/2