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

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

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

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

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

Decimal

0.15907 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

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

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

= [\frac{1}{4} (x^{4} ln (x) - \frac{x ^{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)x^3ln(x) ?
The integral from 1 to sqrt(2 of)x^3ln(x) is (2ln(2)-1)/4+1/16