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

The integral from 3 to 8 of x^3ln(3x) is 1024ln(24)-4015/16-81/2 ln(3)

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

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

1024 ln (24) - \frac{4015}{16} - \frac{81}{2} ln (3)

Decimal Notation

2958.89582 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

= [\frac{1}{4} x^{4} ln (3 x) - \frac{x ^{4}}{16}]_{3}^{8}

Compute the boundaries:

1024 ln (24) - \frac{4015}{16} - \frac{81}{2} ln (3)

= 1024 ln (24) - \frac{4015}{16} - \frac{81}{2} ln (3)

y^{^{''}} - 8 y^{^{'}} + 20 y = 200 x^{2} - 39 x e^{x}

\int_{0}^{8} x e^{- x} d x

\frac{d}{d x} (- \frac{x}{x ^{2} + y ^{2}})

\int \frac{1}{2 - 3 x ^{2}} d x

x \to π - lim (cot (x))

What is the integral from 3 to 8 of x^3ln(3x) ?
The integral from 3 to 8 of x^3ln(3x) is 1024ln(24)-4015/16-81/2 ln(3)