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

The integral from 1 to 3 of x^3(ln(x))^2 is 5/2+81/4 ln^2(3)-81/8 ln(3)

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

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

\frac{5}{2} + \frac{81}{4} ln^{2} (3) - \frac{81}{8} ln (3)

Decimal Notation

15.81726 \dots

Solution steps

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

Simplify

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

Apply Integration By Parts

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

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

\int \frac{1}{2} x^{3} ln (x) d x = \frac{1}{2} (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16})

= [\frac{1}{4} x^{4} ln^{2} (x) - \frac{1}{2} (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16})]_{1}^{3}

Compute the boundaries:

\frac{5}{2} + \frac{81}{4} ln^{2} (3) - \frac{81}{8} ln (3)

= \frac{5}{2} + \frac{81}{4} ln^{2} (3) - \frac{81}{8} ln (3)

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What is the integral from 1 to 3 of x^3(ln(x))^2 ?
The integral from 1 to 3 of x^3(ln(x))^2 is 5/2+81/4 ln^2(3)-81/8 ln(3)