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

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

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

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

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

Decimal

1.94074 \dots

Solution steps

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

Apply Integration By Parts

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

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

= [\frac{2}{3} t^{\frac{3}{2}} ln (t) - \frac{4}{9} t^{\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}

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