What is the integral from 0 to 1/3 of xln(3x) ?

The integral from 0 to 1/3 of xln(3x) is -1/36

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integral from 0 to 1/3 of xln(3x)

\int_{0}^{\frac{1}{3}} x ln (3 x) d x

- \frac{1}{36}

Decimal Notation

- 0.02777 \dots

Solution steps

\int_{0}^{\frac{1}{3}} x ln (3 x) d x

Apply Integration By Parts

: [\frac{1}{2} (x^{2} ln (3 x) - 2 \cdot \int \frac{x}{2} d x)]_{0}^{\frac{1}{3}}

= [\frac{1}{2} (x^{2} ln (3 x) - 2 \cdot \int \frac{x}{2} d x)]_{0}^{\frac{1}{3}}

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

= [\frac{1}{2} (x^{2} ln (3 x) - 2 \cdot \frac{x ^{2}}{4})]_{0}^{\frac{1}{3}}

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

= [\frac{1}{2} (x^{2} ln (3 x) - \frac{x ^{2}}{2})]_{0}^{\frac{1}{3}}

Compute the boundaries:

- \frac{1}{36}

= - \frac{1}{36}

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What is the integral from 0 to 1/3 of xln(3x) ?
The integral from 0 to 1/3 of xln(3x) is -1/36