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

The integral from 1 to sqrt(2 of)t^3ln(t) is (2ln(2)-1)/4+1/16

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integral from 1 to sqrt(2 of)t^3ln(t)

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

\frac{2 ln ( 2 ) - 1}{4} + \frac{1}{16}

Solution steps

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

Apply Integration By Parts

: [\frac{1}{4} (t^{4} ln (t) - 4 \cdot \int \frac{t ^{3}}{4} d t)]_{1}^{2}

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

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

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

4 \cdot \frac{t ^{4}}{16} = \frac{t ^{4}}{4}

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

Compute the boundaries:

\frac{2 ln ( 2 ) - 1}{4} + \frac{1}{16}

= \frac{2 ln ( 2 ) - 1}{4} + \frac{1}{16}

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What is the integral from 1 to sqrt(2 of)t^3ln(t) ?
The integral from 1 to sqrt(2 of)t^3ln(t) is (2ln(2)-1)/4+1/16