What is the integral from 0 to 1/2 of xln(2x) ?

The integral from 0 to 1/2 of xln(2x) is -1/16

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integral from 0 to 1/2 of xln(2x)

\int_{0}^{\frac{1}{2}} x ln (2 x) d x

- \frac{1}{16}

Decimal Notation

- 0.0625

Solution steps

\int_{0}^{\frac{1}{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)]_{0}^{\frac{1}{2}}

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

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

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

Compute the boundaries:

- \frac{1}{16}

= - \frac{1}{16}

\frac{d}{d x} (14 x^{\frac{1}{7}})

n = 0 \sum \infty \frac{( 3 x - 2 ) ^{n}}{n}

y^{^{''}} - 4 y = (x^{2} - 5) sin (2 x)

n = 1 \sum \infty - \frac{5}{n}

\int 7 x + 6 d x

What is the integral from 0 to 1/2 of xln(2x) ?
The integral from 0 to 1/2 of xln(2x) is -1/16