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

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

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integral from 0 to 2 of xln(8x)

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

8 ln (2) - 1

Decimal

4.54517 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

8 ln (2) - 1

= 8 ln (2) - 1

\int_{0}^{2} x^{2} - 3 x + 2 d x

\int_{0}^{π} 3 e^{x} - 2 sin (x) d x

\int_{0}^{2} x ∣ x - 1 ∣ d x

\int_{0}^{1} e^{x} 1 - x d x

\int_{0}^{2} e^{x^{2} - x} d x

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