What is the integral from 1 to 3 of 3x^2ln(2x^3) ?

The integral from 1 to 3 of 3x^2ln(2x^3) is 26ln(2)+81ln(3)-26

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integral from 1 to 3 of 3x^2ln(2x^3)

\int_{1}^{3} 3 x^{2} ln (2 x^{3}) d x

26 ln (2) + 81 ln (3) - 26

Decimal

81.00942 \dots

Solution steps

\int_{1}^{3} 3 x^{2} ln (2 x^{3}) d x

Take the constant out:

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

= 3 \cdot \int_{1}^{3} x^{2} ln (2 x^{3}) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Compute the boundaries:

\frac{26 ln ( 2 ) + 81 ln ( 3 ) - 26}{3}

= 3 \cdot \frac{26 ln ( 2 ) + 81 ln ( 3 ) - 26}{3}

Simplify

= 26 ln (2) + 81 ln (3) - 26

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What is the integral from 1 to 3 of 3x^2ln(2x^3) ?
The integral from 1 to 3 of 3x^2ln(2x^3) is 26ln(2)+81ln(3)-26