What is the integral from 1/2 to 3 of 16xln(2x) ?

The integral from 1/2 to 3 of 16xln(2x) is -35+72ln(6)

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integral from 1/2 to 3 of 16xln(2x)

\int_{\frac{1}{2}}^{3} 16 x ln (2 x) d x

- 35 + 72 ln (6)

Decimal

94.00668 \dots

Solution steps

\int_{\frac{1}{2}}^{3} 16 x ln (2 x) d x

Take the constant out:

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

= 16 \cdot \int_{\frac{1}{2}}^{3} x ln (2 x) d x

Apply Integration By Parts

= 16 [\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}

= 16 [\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)

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

Simplify

= - 35 + 72 ln (6)

What is the integral from 1/2 to 3 of 16xln(2x) ?
The integral from 1/2 to 3 of 16xln(2x) is -35+72ln(6)