What is the integral from 1/2 to 7 of 8xln(2x) ?

The integral from 1/2 to 7 of 8xln(2x) is 8(-195/16+49/2 ln(14))

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integral from 1/2 to 7 of 8xln(2x)

\int_{\frac{1}{2}}^{7} 8 x ln (2 x) d x

8 (- \frac{195}{16} + \frac{49}{2} ln (14))

Decimal Notation

419.75523 \dots

Solution steps

\int_{\frac{1}{2}}^{7} 8 x ln (2 x) d x

Take the constant out:

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

= 8 \cdot \int_{\frac{1}{2}}^{7} x ln (2 x) d x

Apply Integration By Parts

: [\frac{1}{2} x^{2} ln (2 x) - \int \frac{x}{2} d x]_{\frac{1}{2}}^{7}

= 8 [\frac{1}{2} x^{2} ln (2 x) - \int \frac{x}{2} d x]_{\frac{1}{2}}^{7}

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

= 8 [\frac{1}{2} x^{2} ln (2 x) - \frac{x ^{2}}{4}]_{\frac{1}{2}}^{7}

Compute the boundaries:

- \frac{195}{16} + \frac{49}{2} ln (14)

= 8 (- \frac{195}{16} + \frac{49}{2} ln (14))

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What is the integral from 1/2 to 7 of 8xln(2x) ?
The integral from 1/2 to 7 of 8xln(2x) is 8(-195/16+49/2 ln(14))