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}

\frac{\partial}{\partial y} (π x x^{2} + y^{2})

\frac{- 1}{( 2 x + 1 ) 2 x + 1}

\frac{d}{d x} (1 - 3 x)

\frac{d}{d x} (x^{3} + 3 x^{2} - 9 x + 1)

\frac{d}{d x} (e^{8 - x})

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