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

The integral from 1 to 3 of sqrt(x)(ln(x)) is 2sqrt(3)ln(3)-4/(sqrt(3))+4/9

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

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

23 ln (3) - \frac{4}{3} + \frac{4}{9}

Decimal Notation

1.94074 \dots

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{2}{3} x^{\frac{1}{2}} d x = \frac{4}{9} x^{\frac{3}{2}}

= [\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{2}}]_{1}^{3}

Compute the boundaries:

23 ln (3) - \frac{4}{3} + \frac{4}{9}

= 23 ln (3) - \frac{4}{3} + \frac{4}{9}

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What is the integral from 1 to 3 of sqrt(x)(ln(x)) ?
The integral from 1 to 3 of sqrt(x)(ln(x)) is 2sqrt(3)ln(3)-4/(sqrt(3))+4/9