What is the integral from 0 to e of x*ln(x) ?

The integral from 0 to e of x*ln(x) is (e^2)/4

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integral from 0 to e of x*ln(x)

\int_{0}^{e} x \cdot ln (x) d x

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

Decimal Notation

1.84726 \dots

Solution steps

\int_{0}^{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)]_{0}^{e}

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

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

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

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

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

Compute the boundaries:

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

= \frac{e ^{2}}{4}

\int 0.2 x + 8 d x

e^{x}, e^{- x}, x = - 2, x = 1

\int e^{- 0.125 x} d x

\frac{\partial}{\partial y} (e^{x y^{2}} + 4 z y^{3})

\frac{d}{d x} (\frac{( x + 2 ) \cdot ( x + 4 )}{x ^{2} - 4 x + 20})

What is the integral from 0 to e of x*ln(x) ?
The integral from 0 to e of x*ln(x) is (e^2)/4