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Beliebt Rechnen >

integral from 1 to e of x(ln(x))^2

Lösung

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

Lösung

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

+1

Dezimale

1.59726 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int x ln (x) d x = \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}

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

Vereinfache

[\frac{1}{2} (x^{2} ln^{2} (x) - 2 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}))]_{1}^{e} : [\frac{1}{4} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2})]_{1}^{e}

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

Berechne die Grenzen:

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

= \frac{e ^{2}}{4} - \frac{1}{4}

Vereinfache

= \frac{e ^{2} - 1}{4}

Graph

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