integral from 1 to e of 8xln(x)

Lösung

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

2 (e^{2} + 1)

Dezimale

16.77811 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

= 8 [\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}

= 8 [\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}

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

Berechne die Grenzen:

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

= 8 (\frac{e ^{2}}{4} + \frac{1}{4})

Vereinfache

= 2 (e^{2} + 1)

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