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

Lösung

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

\frac{5 e ^{3} - 2}{27}

Dezimale

3.64546 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

= [\frac{1}{3} (x^{3} ln^{2} (x) - 3 \cdot \int \frac{2 x ^{2} ln ( x )}{3} d x)]_{1}^{e}

\int \frac{2 x ^{2} ln ( x )}{3} d x = \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9})

= [\frac{1}{3} (x^{3} ln^{2} (x) - 3 \cdot \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}))]_{1}^{e}

Vereinfache

[\frac{1}{3} (x^{3} ln^{2} (x) - 3 \cdot \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}))]_{1}^{e} : [\frac{1}{3} (x^{3} ln^{2} (x) - 2 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}))]_{1}^{e}

= [\frac{1}{3} (x^{3} ln^{2} (x) - 2 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}))]_{1}^{e}

Berechne die Grenzen:

\frac{5 e ^{3}}{27} - \frac{2}{27}

= \frac{5 e ^{3}}{27} - \frac{2}{27}

Vereinfache

= \frac{5 e ^{3} - 2}{27}

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