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

The integral from 1/2 to 1 of x^2ln(x) is -7/72+1/24 ln(2)

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integral from 1/2 to 1 of x^2ln(x)

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

- \frac{7}{72} + \frac{1}{24} ln (2)

Decimal Notation

- 0.06834 \dots

Solution steps

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

Apply Integration By Parts

: [\frac{1}{3} x^{3} ln (x) - \int \frac{x ^{2}}{3} d x]_{\frac{1}{2}}^{1}

= [\frac{1}{3} x^{3} ln (x) - \int \frac{x ^{2}}{3} d x]_{\frac{1}{2}}^{1}

\int \frac{x ^{2}}{3} d x = \frac{x ^{3}}{9}

= [\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}]_{\frac{1}{2}}^{1}

Compute the boundaries:

- \frac{7}{72} + \frac{1}{24} ln (2)

= - \frac{7}{72} + \frac{1}{24} ln (2)

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What is the integral from 1/2 to 1 of x^2ln(x) ?
The integral from 1/2 to 1 of x^2ln(x) is -7/72+1/24 ln(2)