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

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

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integral from 0 to 2 of z^2ln(z)

\int_{0}^{2} z^{2} ln (z) d z

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

Decimal

0.95950 \dots

Solution steps

\int_{0}^{2} z^{2} ln (z) d z

Apply Integration By Parts

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

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

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

= [\frac{1}{3} z^{3} ln (z) - \frac{z ^{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 z^2ln(z) ?
The integral from 0 to 2 of z^2ln(z) is 8/3 ln(2)-8/9