What is the integral from 2 to 5 of x^2ln(x) ?

The integral from 2 to 5 of x^2ln(x) is -13+(125ln(5)-8ln(2))/3

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integral from 2 to 5 of x^2ln(x)

\int_{2}^{5} x^{2} ln (x) d x

- 13 + \frac{125 ln ( 5 ) - 8 ln ( 2 )}{3}

Decimal

52.21152 \dots

Solution steps

\int_{2}^{5} x^{2} ln (x) d x

Apply Integration By Parts

: [\frac{1}{3} x^{3} ln (x) - \int \frac{x ^{2}}{3} d x]_{2}^{5}

= [\frac{1}{3} x^{3} ln (x) - \int \frac{x ^{2}}{3} d x]_{2}^{5}

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

= [\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}]_{2}^{5}

Compute the boundaries:

- 13 + \frac{125 ln ( 5 ) - 8 ln ( 2 )}{3}

= - 13 + \frac{125 ln ( 5 ) - 8 ln ( 2 )}{3}

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What is the integral from 2 to 5 of x^2ln(x) ?
The integral from 2 to 5 of x^2ln(x) is -13+(125ln(5)-8ln(2))/3