What is the integral from 1 to 4 of-1sqrt(t)ln(t) ?

The integral from 1 to 4 of-1sqrt(t)ln(t) is 28/9-32/3 ln(2)

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integral from 1 to 4 of-1sqrt(t)ln(t)

\int_{1}^{4} - 1 t ln (t) d t

\frac{28}{9} - \frac{32}{3} ln (2)

Decimal

- 4.28245 \dots

Solution steps

\int_{1}^{4} - 1 \cdot t ln (t) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int_{1}^{4} 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}^{4}

\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}^{4}

Compute the boundaries:

- \frac{28}{9} + \frac{32}{3} ln (2)

= - (- \frac{28}{9} + \frac{32}{3} ln (2))

Simplify

= \frac{28}{9} - \frac{32}{3} ln (2)

What is the integral from 1 to 4 of-1sqrt(t)ln(t) ?
The integral from 1 to 4 of-1sqrt(t)ln(t) is 28/9-32/3 ln(2)