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

The integral from 3 to 8 of t^3ln(t) is 3072ln(2)-4015/16-81/4 ln(3)

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

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

3072 ln (2) - \frac{4015}{16} - \frac{81}{4} ln (3)

Decimal Notation

1856.16373 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

3072 ln (2) - \frac{4015}{16} - \frac{81}{4} ln (3)

= 3072 ln (2) - \frac{4015}{16} - \frac{81}{4} ln (3)

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What is the integral from 3 to 8 of t^3ln(t) ?
The integral from 3 to 8 of t^3ln(t) is 3072ln(2)-4015/16-81/4 ln(3)