What is the integral from 0 to 1 of 8xln(2x) ?

The integral from 0 to 1 of 8xln(2x) is 8(1/2 ln(2)-1/4)

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integral from 0 to 1 of 8xln(2x)

\int_{0}^{1} 8 x ln (2 x) d x

8 (\frac{1}{2} ln (2) - \frac{1}{4})

Decimal

0.77258 \dots

Solution steps

\int_{0}^{1} 8 x ln (2 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8 \cdot \int_{0}^{1} x ln (2 x) d x

Apply Integration By Parts

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

= 8 [\frac{1}{2} x^{2} ln (2 x) - \int \frac{x}{2} d x]_{0}^{1}

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

= 8 [\frac{1}{2} x^{2} ln (2 x) - \frac{x ^{2}}{4}]_{0}^{1}

Compute the boundaries:

\frac{1}{2} ln (2) - \frac{1}{4}

= 8 (\frac{1}{2} ln (2) - \frac{1}{4})

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What is the integral from 0 to 1 of 8xln(2x) ?
The integral from 0 to 1 of 8xln(2x) is 8(1/2 ln(2)-1/4)