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

The integral from 2 to 8 of x^3ln(2x) is 4088ln(2)-255

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

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

4088 ln (2) - 255

Decimal

2578.58567 \dots

Solution steps

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

Apply Integration By Parts

= [\frac{1}{4} x^{4} ln (2 x) - \int \frac{x ^{3}}{4} d x]_{2}^{8}

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

= [\frac{1}{4} x^{4} ln (2 x) - \frac{x ^{4}}{16}]_{2}^{8}

Compute the boundaries:

4088 ln (2) - 255

= 4088 ln (2) - 255

What is the integral from 2 to 8 of x^3ln(2x) ?
The integral from 2 to 8 of x^3ln(2x) is 4088ln(2)-255