What is the integral from 1 to e of 10xln(x) ?

The integral from 1 to e of 10xln(x) is (5(e^2+1))/2

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integral from 1 to e of 10xln(x)

\int_{1}^{e} 10 x ln (x) d x

\frac{5 ( e ^{2} + 1 )}{2}

Decimal

20.97264 \dots

Solution steps

\int_{1}^{e} 10 x ln (x) d x

Take the constant out:

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

= 10 \cdot \int_{1}^{e} x ln (x) d x

Apply Integration By Parts

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

= 10 [\frac{1}{2} (x^{2} ln (x) - 2 \cdot \int \frac{x}{2} d x)]_{1}^{e}

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

= 10 [\frac{1}{2} (x^{2} ln (x) - 2 \cdot \frac{x ^{2}}{4})]_{1}^{e}

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

= 10 [\frac{1}{2} (x^{2} ln (x) - \frac{x ^{2}}{2})]_{1}^{e}

Compute the boundaries:

\frac{e ^{2}}{4} + \frac{1}{4}

= 10 (\frac{e ^{2}}{4} + \frac{1}{4})

Simplify

= \frac{5 ( e ^{2} + 1 )}{2}

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What is the integral from 1 to e of 10xln(x) ?
The integral from 1 to e of 10xln(x) is (5(e^2+1))/2