What is the integral from 0 to 1/5 of xln(5x) ?

The integral from 0 to 1/5 of xln(5x) is -1/100

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integral from 0 to 1/5 of xln(5x)

\int_{0}^{\frac{1}{5}} x ln (5 x) d x

- \frac{1}{100}

Decimal Notation

- 0.01

Solution steps

\int_{0}^{\frac{1}{5}} x ln (5 x) d x

Apply Integration By Parts

: [\frac{1}{2} (x^{2} ln (5 x) - 2 \cdot \int \frac{x}{2} d x)]_{0}^{\frac{1}{5}}

= [\frac{1}{2} (x^{2} ln (5 x) - 2 \cdot \int \frac{x}{2} d x)]_{0}^{\frac{1}{5}}

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

= [\frac{1}{2} (x^{2} ln (5 x) - 2 \cdot \frac{x ^{2}}{4})]_{0}^{\frac{1}{5}}

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

= [\frac{1}{2} (x^{2} ln (5 x) - \frac{x ^{2}}{2})]_{0}^{\frac{1}{5}}

Compute the boundaries:

- \frac{1}{100}

= - \frac{1}{100}

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What is the integral from 0 to 1/5 of xln(5x) ?
The integral from 0 to 1/5 of xln(5x) is -1/100