integral from 1 to e of x^4(ln(x))^2

Lösung

\int_{1}^{e} x^{4} (ln (x))^{2} d x

\frac{17 e ^{5} - 2}{125}

Dezimale

20.16818 \dots

Schritte zur Lösung

\int_{1}^{e} x^{4} (ln (x))^{2} d x

Wende die partielle Integration an

= [\frac{1}{5} (x^{5} ln^{2} (x) - 5 \cdot \int \frac{2 x ^{4} ln ( x )}{5} d x)]_{1}^{e}

\int \frac{2 x ^{4} ln ( x )}{5} d x = \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25})

= [\frac{1}{5} (x^{5} ln^{2} (x) - 5 \cdot \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}))]_{1}^{e}

Vereinfache

[\frac{1}{5} (x^{5} ln^{2} (x) - 5 \cdot \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}))]_{1}^{e} : [\frac{1}{5} (x^{5} ln^{2} (x) - 2 (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}))]_{1}^{e}

= [\frac{1}{5} (x^{5} ln^{2} (x) - 2 (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}))]_{1}^{e}

Berechne die Grenzen:

\frac{17 e ^{5}}{125} - \frac{2}{125}

= \frac{17 e ^{5}}{125} - \frac{2}{125}

Vereinfache

= \frac{17 e ^{5} - 2}{125}

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