What is the integral from 0 to 1 of sqrt(x)*ln(x) ?

The integral from 0 to 1 of sqrt(x)*ln(x) is -4/9

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integral from 0 to 1 of sqrt(x)*ln(x)

\int_{0}^{1} x \cdot ln (x) d x

- \frac{4}{9}

Decimal

- 0.44444 \dots

Solution steps

\int_{0}^{1} 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]_{0}^{1}

\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}}]_{0}^{1}

Compute the boundaries:

- \frac{4}{9}

= - \frac{4}{9}

What is the integral from 0 to 1 of sqrt(x)*ln(x) ?
The integral from 0 to 1 of sqrt(x)*ln(x) is -4/9