What is the integral from 1/2 to e/2 of xln(2x) ?

The integral from 1/2 to e/2 of xln(2x) is (e^2+1)/(16)

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integral from 1/2 to e/2 of xln(2x)

\int_{\frac{1}{2}}^{\frac{e}{2}} x ln (2 x) d x

\frac{e ^{2} + 1}{16}

Solution steps

\int_{\frac{1}{2}}^{\frac{e}{2}} 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)]_{\frac{1}{2}}^{\frac{e}{2}}

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

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

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

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

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

Compute the boundaries:

\frac{e ^{2}}{16} + \frac{1}{16}

= \frac{e ^{2}}{16} + \frac{1}{16}

Simplify

= \frac{e ^{2} + 1}{16}

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What is the integral from 1/2 to e/2 of xln(2x) ?
The integral from 1/2 to e/2 of xln(2x) is (e^2+1)/(16)