What is the integral from 0 to 2 of 1x^2ln(x) ?

The integral from 0 to 2 of 1x^2ln(x) is 8/3 ln(2)-8/9

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integral from 0 to 2 of 1x^2ln(x)

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

\frac{8}{3} ln (2) - \frac{8}{9}

Decimal

0.95950 \dots

Solution steps

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

Simplify

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

Apply Integration By Parts

: [\frac{1}{3} x^{3} ln (x) - \int \frac{x ^{2}}{3} d x]_{0}^{2}

= [\frac{1}{3} x^{3} ln (x) - \int \frac{x ^{2}}{3} d x]_{0}^{2}

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

= [\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}]_{0}^{2}

Compute the boundaries:

\frac{8}{3} ln (2) - \frac{8}{9}

= \frac{8}{3} ln (2) - \frac{8}{9}

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What is the integral from 0 to 2 of 1x^2ln(x) ?
The integral from 0 to 2 of 1x^2ln(x) is 8/3 ln(2)-8/9