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

The integral from 1/2 to 1 of xln(x) is -3/16+1/8 ln(2)

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integral from 1/2 to 1 of xln(x)

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

- \frac{3}{16} + \frac{1}{8} ln (2)

Decimal Notation

- 0.10085 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

- \frac{3}{16} + \frac{1}{8} ln (2)

= - \frac{3}{16} + \frac{1}{8} ln (2)

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What is the integral from 1/2 to 1 of xln(x) ?
The integral from 1/2 to 1 of xln(x) is -3/16+1/8 ln(2)