What is the integral from 1/3 to 5 of 10xln(3x) ?

The integral from 1/3 to 5 of 10xln(3x) is 10(-56/9+25/2 ln(15))

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integral from 1/3 to 5 of 10xln(3x)

\int_{\frac{1}{3}}^{5} 10 x ln (3 x) d x

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

Decimal

276.28405 \dots

Solution steps

\int_{\frac{1}{3}}^{5} 10 x ln (3 x) d x

Take the constant out:

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

= 10 \cdot \int_{\frac{1}{3}}^{5} x ln (3 x) d x

Apply Integration By Parts

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

= 10 [\frac{1}{2} x^{2} ln (3 x) - \int \frac{x}{2} d x]_{\frac{1}{3}}^{5}

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

= 10 [\frac{1}{2} x^{2} ln (3 x) - \frac{x ^{2}}{4}]_{\frac{1}{3}}^{5}

Compute the boundaries:

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

= 10 (- \frac{56}{9} + \frac{25}{2} ln (15))

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What is the integral from 1/3 to 5 of 10xln(3x) ?
The integral from 1/3 to 5 of 10xln(3x) is 10(-56/9+25/2 ln(15))