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

The integral from 1 to 2 of x^2ln(2x) is 5ln(2)-7/9

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

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

5 ln (2) - \frac{7}{9}

Decimal

2.68795 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

5 ln (2) - \frac{7}{9}

= 5 ln (2) - \frac{7}{9}

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What is the integral from 1 to 2 of x^2ln(2x) ?
The integral from 1 to 2 of x^2ln(2x) is 5ln(2)-7/9