What is the integral from 1 to 8 of sqrt(x)ln(x) ?

The integral from 1 to 8 of sqrt(x)ln(x) is 32sqrt(2)ln(2)+(-64sqrt(2)+4)/9

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integral from 1 to 8 of sqrt(x)ln(x)

\int_{1}^{8} x ln (x) d x

322 ln (2) + \frac{- 64 2 + 4}{9}

Decimal

21.75607 \dots

Solution steps

\int_{1}^{8} 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}^{8}

= [\frac{2}{3} x^{\frac{3}{2}} ln (x) - \int \frac{2}{3} x^{\frac{1}{2}} d x]_{1}^{8}

\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}^{8}

Compute the boundaries:

322 ln (2) - \frac{64 2}{9} + \frac{4}{9}

= 322 ln (2) - \frac{64 2}{9} + \frac{4}{9}

Simplify

= 322 ln (2) + \frac{- 64 2 + 4}{9}

x \to 0 lim (\frac{∣ x - 1 ∣}{x - 1})

f (x) = sin^{2} (3 x + 2)

(e^{2 x} + e^{- x})^{^{'}}

y = - 3 - 5 x^{2}

t cos (3 t)

What is the integral from 1 to 8 of sqrt(x)ln(x) ?
The integral from 1 to 8 of sqrt(x)ln(x) is 32sqrt(2)ln(2)+(-64sqrt(2)+4)/9