What is the integral from 0 to 1/13 of xln(13x) ?

The integral from 0 to 1/13 of xln(13x) is -1/676

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integral from 0 to 1/13 of xln(13x)

\int_{0}^{\frac{1}{13}} x ln (13 x) d x

- \frac{1}{676}

Decimal

- 0.00147 \dots

Solution steps

\int_{0}^{\frac{1}{13}} x ln (13 x) d x

Apply Integration By Parts

= [\frac{1}{2} (x^{2} ln (13 x) - 2 \cdot \int \frac{x}{2} d x)]_{0}^{\frac{1}{13}}

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

= [\frac{1}{2} (x^{2} ln (13 x) - 2 \cdot \frac{x ^{2}}{4})]_{0}^{\frac{1}{13}}

2 \cdot \frac{x ^{2}}{4} = \frac{x ^{2}}{2}

= [\frac{1}{2} (x^{2} ln (13 x) - \frac{x ^{2}}{2})]_{0}^{\frac{1}{13}}

Compute the boundaries:

- \frac{1}{676}

= - \frac{1}{676}

What is the integral from 0 to 1/13 of xln(13x) ?
The integral from 0 to 1/13 of xln(13x) is -1/676