What is the integral from 3 to 8 of 2x^2ln(x-2) ?

The integral from 3 to 8 of 2x^2ln(x-2) is (2(1512ln(6)-710))/9

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integral from 3 to 8 of 2x^2ln(x-2)

\int_{3}^{8} 2 x^{2} ln (x - 2) d x

\frac{2 ( 1512 ln ( 6 ) - 710 )}{9}

Decimal

444.25340 \dots

Solution steps

\int_{3}^{8} 2 x^{2} ln (x - 2) d x

Take the constant out:

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

= 2 \cdot \int_{3}^{8} x^{2} ln (x - 2) d x

Apply Integration By Parts

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

\int \frac{x ^{3}}{3 ( x - 2 )} d x = \frac{1}{9} ((x - 2)^{3} + 9 (x - 2)^{2} + 36 (x - 2) + 24 ln ∣ x - 2 ∣)

= 2 [\frac{1}{3} x^{3} ln (x - 2) - \frac{1}{9} ((x - 2)^{3} + 9 (x - 2)^{2} + 36 (x - 2) + 24 ln ∣ x - 2 ∣)]_{3}^{8}

Compute the boundaries:

\frac{1512 ln ( 6 ) - 756}{9} + \frac{46}{9}

= 2 (\frac{1512 ln ( 6 ) - 756}{9} + \frac{46}{9})

Simplify

= \frac{2 ( 1512 ln ( 6 ) - 710 )}{9}

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What is the integral from 3 to 8 of 2x^2ln(x-2) ?
The integral from 3 to 8 of 2x^2ln(x-2) is (2(1512ln(6)-710))/9