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

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

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

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

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

Decimal

0.32461 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

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

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

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