What is the integral from 1/5 to 3 of 6xln(5x) ?

The integral from 1/5 to 3 of 6xln(5x) is 6(-56/25+9/2 ln(15))

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integral from 1/5 to 3 of 6xln(5x)

\int_{\frac{1}{5}}^{3} 6 x ln (5 x) d x

6 (- \frac{56}{25} + \frac{9}{2} ln (15))

Decimal

59.67735 \dots

Solution steps

\int_{\frac{1}{5}}^{3} 6 x ln (5 x) d x

Take the constant out:

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

= 6 \cdot \int_{\frac{1}{5}}^{3} x ln (5 x) d x

Apply Integration By Parts

= 6 [\frac{1}{2} x^{2} ln (5 x) - \int \frac{x}{2} d x]_{\frac{1}{5}}^{3}

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

= 6 [\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4}]_{\frac{1}{5}}^{3}

Compute the boundaries:

- \frac{56}{25} + \frac{9}{2} ln (15)

= 6 (- \frac{56}{25} + \frac{9}{2} ln (15))

What is the integral from 1/5 to 3 of 6xln(5x) ?
The integral from 1/5 to 3 of 6xln(5x) is 6(-56/25+9/2 ln(15))