What is the integral from 2 to 6 of 6x^2ln(3x) ?

The integral from 2 to 6 of 6x^2ln(3x) is 6(72ln(18)-208/9-8/3 ln(6))

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integral from 2 to 6 of 6x^2ln(3x)

\int_{2}^{6} 6 x^{2} ln (3 x) d x

6 (72 ln (18) - \frac{208}{9} - \frac{8}{3} ln (6))

Decimal

1081.30578 \dots

Solution steps

\int_{2}^{6} 6 x^{2} ln (3 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int_{2}^{6} x^{2} ln (3 x) d x

Apply Integration By Parts

= 6 [\frac{1}{3} x^{3} ln (3 x) - \int \frac{x ^{2}}{3} d x]_{2}^{6}

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

= 6 [\frac{1}{3} x^{3} ln (3 x) - \frac{x ^{3}}{9}]_{2}^{6}

Compute the boundaries:

72 ln (18) - \frac{208}{9} - \frac{8}{3} ln (6)

= 6 (72 ln (18) - \frac{208}{9} - \frac{8}{3} ln (6))

What is the integral from 2 to 6 of 6x^2ln(3x) ?
The integral from 2 to 6 of 6x^2ln(3x) is 6(72ln(18)-208/9-8/3 ln(6))