What is the integral from 0 to 1/2 of xln(x) ?

The integral from 0 to 1/2 of xln(x) is 1/2 (-1/4 ln(2)-1/8)

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integral from 0 to 1/2 of xln(x)

\int_{0}^{\frac{1}{2}} x ln (x) d x

\frac{1}{2} (- \frac{1}{4} ln (2) - \frac{1}{8})

Decimal

- 0.14914 \dots

Solution steps

\int_{0}^{\frac{1}{2}} x ln (x) d x

Apply Integration By Parts

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

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

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

2 \cdot \frac{x ^{2}}{4} = \frac{x ^{2}}{2}

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

Compute the boundaries:

\frac{1}{2} (- \frac{1}{4} ln (2) - \frac{1}{8})

= \frac{1}{2} (- \frac{1}{4} ln (2) - \frac{1}{8})

What is the integral from 0 to 1/2 of xln(x) ?
The integral from 0 to 1/2 of xln(x) is 1/2 (-1/4 ln(2)-1/8)