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

The integral from e to e^2 of xln(x) is (3e^4-e^2)/4

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integral from e to e^2 of xln(x)

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

\frac{3 e ^{4} - e ^{2}}{4}

Solution steps

\int_{e}^{e^{2}} 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}^{e^{2}}

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

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

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

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

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

Compute the boundaries:

\frac{3 e ^{4} - e ^{2}}{4}

= \frac{3 e ^{4} - e ^{2}}{4}

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What is the integral from e to e^2 of xln(x) ?
The integral from e to e^2 of xln(x) is (3e^4-e^2)/4