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

The integral from 0 to 1 of-xln(x) is 1/4

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integral from 0 to 1 of-xln(x)

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

\frac{1}{4}

Decimal

0.25

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= - [\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}]_{0}^{1}

Compute the boundaries:

- \frac{1}{4}

= - (- \frac{1}{4})

Simplify

= \frac{1}{4}

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What is the integral from 0 to 1 of-xln(x) ?
The integral from 0 to 1 of-xln(x) is 1/4