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

The integral from e to 2e of xln(x) is (8e^2ln(2e)-5e^2)/4

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integral from e to 2e of xln(x)

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

\frac{8 e ^{2} ln ( 2 e ) - 5 e ^{2}}{4}

Decimal

15.78519 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

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

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

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

Compute the boundaries:

\frac{8 e ^{2} ln ( 2 e ) - 5 e ^{2}}{4}

= \frac{8 e ^{2} ln ( 2 e ) - 5 e ^{2}}{4}

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What is the integral from e to 2e of xln(x) ?
The integral from e to 2e of xln(x) is (8e^2ln(2e)-5e^2)/4