What is the integral from 1/7 to 3 of 12xln(7x) ?

The integral from 1/7 to 3 of 12xln(7x) is 12(-110/49+9/2 ln(21))

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integral from 1/7 to 3 of 12xln(7x)

\int_{\frac{1}{7}}^{3} 12 x ln (7 x) d x

12 (- \frac{110}{49} + \frac{9}{2} ln (21))

Decimal Notation

137.46543 \dots

Solution steps

\int_{\frac{1}{7}}^{3} 12 x ln (7 x) d x

Take the constant out:

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

= 12 \cdot \int_{\frac{1}{7}}^{3} x ln (7 x) d x

Apply Integration By Parts

: [\frac{1}{2} x^{2} ln (7 x) - \int \frac{x}{2} d x]_{\frac{1}{7}}^{3}

= 12 [\frac{1}{2} x^{2} ln (7 x) - \int \frac{x}{2} d x]_{\frac{1}{7}}^{3}

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

= 12 [\frac{1}{2} x^{2} ln (7 x) - \frac{x ^{2}}{4}]_{\frac{1}{7}}^{3}

Compute the boundaries:

- \frac{110}{49} + \frac{9}{2} ln (21)

= 12 (- \frac{110}{49} + \frac{9}{2} ln (21))

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What is the integral from 1/7 to 3 of 12xln(7x) ?
The integral from 1/7 to 3 of 12xln(7x) is 12(-110/49+9/2 ln(21))