What is the integral from 1 to 4 of 3x^3ln(3x^3) ?

The integral from 1 to 4 of 3x^3ln(3x^3) is 3(384ln(2)-765/16+255/4 ln(3))

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integral from 1 to 4 of 3x^3ln(3x^3)

\int_{1}^{4} 3 x^{3} ln (3 x^{3}) d x

3 (384 ln (2) - \frac{765}{16} + \frac{255}{4} ln (3))

Decimal

865.17765 \dots

Solution steps

\int_{1}^{4} 3 x^{3} ln (3 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}^{4} x^{3} ln (3 x^{3}) d x

Apply Integration By Parts

= 3 [\frac{1}{4} x^{4} ln (3 x^{3}) - \int \frac{3 x ^{3}}{4} d x]_{1}^{4}

\int \frac{3 x ^{3}}{4} d x = \frac{3 x ^{4}}{16}

= 3 [\frac{1}{4} x^{4} ln (3 x^{3}) - \frac{3 x ^{4}}{16}]_{1}^{4}

Simplify

ln (3 x^{3}) : 3 ln (x) + ln (3)

= 3 [\frac{1}{4} x^{4} (ln (3) + 3 ln (x)) - \frac{3 x ^{4}}{16}]_{1}^{4}

Compute the boundaries:

384 ln (2) - \frac{765}{16} + \frac{255}{4} ln (3)

= 3 (384 ln (2) - \frac{765}{16} + \frac{255}{4} ln (3))

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What is the integral from 1 to 4 of 3x^3ln(3x^3) ?
The integral from 1 to 4 of 3x^3ln(3x^3) is 3(384ln(2)-765/16+255/4 ln(3))