What is the integral from 1 to 8 of x^2ln(x) ?

The integral from 1 to 8 of x^2ln(x) is 512ln(2)-511/9

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integral from 1 to 8 of x^2ln(x)

\int_{1}^{8} x^{2} ln (x) d x

512 ln (2) - \frac{511}{9}

Decimal

298.11357 \dots

Solution steps

\int_{1}^{8} x^{2} ln (x) d x

Apply Integration By Parts

: [\frac{1}{3} x^{3} ln (x) - \int \frac{x ^{2}}{3} d x]_{1}^{8}

= [\frac{1}{3} x^{3} ln (x) - \int \frac{x ^{2}}{3} d x]_{1}^{8}

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

= [\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}]_{1}^{8}

Compute the boundaries:

512 ln (2) - \frac{511}{9}

= 512 ln (2) - \frac{511}{9}

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What is the integral from 1 to 8 of x^2ln(x) ?
The integral from 1 to 8 of x^2ln(x) is 512ln(2)-511/9