What is the integral from 0 to 1/9 of xln(9x) ?

The integral from 0 to 1/9 of xln(9x) is -1/324

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integral from 0 to 1/9 of xln(9x)

\int_{0}^{\frac{1}{9}} x ln (9 x) d x

- \frac{1}{324}

Decimal

- 0.00308 \dots

Solution steps

\int_{0}^{\frac{1}{9}} x ln (9 x) d x

Apply Integration By Parts

= [\frac{1}{2} (x^{2} ln (9 x) - 2 \cdot \int \frac{x}{2} d x)]_{0}^{\frac{1}{9}}

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

= [\frac{1}{2} (x^{2} ln (9 x) - 2 \cdot \frac{x ^{2}}{4})]_{0}^{\frac{1}{9}}

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

= [\frac{1}{2} (x^{2} ln (9 x) - \frac{x ^{2}}{2})]_{0}^{\frac{1}{9}}

Compute the boundaries:

- \frac{1}{324}

= - \frac{1}{324}

What is the integral from 0 to 1/9 of xln(9x) ?
The integral from 0 to 1/9 of xln(9x) is -1/324