What is the integral from 0 to e of xln(2x) ?

The integral from 0 to e of xln(2x) is 1/2 (e^2ln(2e)-(e^2)/2)

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integral from 0 to e of xln(2x)

\int_{0}^{e} x ln (2 x) d x

\frac{1}{2} (e^{2} ln (2 e) - \frac{e ^{2}}{2})

Decimal Notation

4.40811 \dots

Solution steps

\int_{0}^{e} x ln (2 x) d x

Apply Integration By Parts

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

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

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

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

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

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

Compute the boundaries:

\frac{1}{2} (e^{2} ln (2 e) - \frac{e ^{2}}{2})

= \frac{1}{2} (e^{2} ln (2 e) - \frac{e ^{2}}{2})

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What is the integral from 0 to e of xln(2x) ?
The integral from 0 to e of xln(2x) is 1/2 (e^2ln(2e)-(e^2)/2)