What is the integral from 1/2 to 3 of 4xln(2x) ?

The integral from 1/2 to 3 of 4xln(2x) is 4(-35/16+9/2 ln(6))

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integral from 1/2 to 3 of 4xln(2x)

\int_{\frac{1}{2}}^{3} 4 x ln (2 x) d x

4 (- \frac{35}{16} + \frac{9}{2} ln (6))

Decimal

23.50167 \dots

Solution steps

\int_{\frac{1}{2}}^{3} 4 x ln (2 x) d x

Take the constant out:

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

= 4 \cdot \int_{\frac{1}{2}}^{3} x ln (2 x) d x

Apply Integration By Parts

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

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

Compute the boundaries:

- \frac{35}{16} + \frac{9}{2} ln (6)

= 4 (- \frac{35}{16} + \frac{9}{2} ln (6))

What is the integral from 1/2 to 3 of 4xln(2x) ?
The integral from 1/2 to 3 of 4xln(2x) is 4(-35/16+9/2 ln(6))