What is the integral from 1 to 4 of sqrt(x)ln(x/2) ?

The integral from 1 to 4 of sqrt(x)ln(x/2) is 6ln(2)-28/9

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integral from 1 to 4 of sqrt(x)ln(x/2)

\int_{1}^{4} x ln (\frac{x}{2}) d x

6 ln (2) - \frac{28}{9}

Decimal

1.04777 \dots

Solution steps

\int_{1}^{4} x ln (\frac{x}{2}) d x

Apply Integration By Parts

= [\frac{2}{3} x^{\frac{3}{2}} ln (\frac{x}{2}) - \int \frac{2}{3} x^{\frac{1}{2}} d x]_{1}^{4}

\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 (\frac{x}{2}) - \frac{4}{9} x^{\frac{3}{2}}]_{1}^{4}

Compute the boundaries:

6 ln (2) - \frac{28}{9}

= 6 ln (2) - \frac{28}{9}

y^{^{'''}} - 6 y^{^{''}} + 11 y^{^{'}} - 6 y = 0

\int_{- 5}^{1} (2 x + 6) d x

d er i v a t i v e \frac{7}{t}

l a pl a ce t r an s f or m \frac{1}{2} t^{2}

a re a y = 2 x - 1, x = 2, x = 5, y = 0

What is the integral from 1 to 4 of sqrt(x)ln(x/2) ?
The integral from 1 to 4 of sqrt(x)ln(x/2) is 6ln(2)-28/9