integral from 1 to e of x^5ln(x^2)

Lösung

\int_{1}^{e} x^{5} ln (x^{2}) d x

\frac{5 e ^{6} + 1}{18}

Dezimale

112.11910 \dots

Schritte zur Lösung

\int_{1}^{e} x^{5} ln (x^{2}) d x

Wende die partielle Integration an

= [\frac{1}{6} (x^{6} ln (x^{2}) - 6 \cdot \int \frac{x ^{5}}{3} d x)]_{1}^{e}

\int \frac{x ^{5}}{3} d x = \frac{x ^{6}}{18}

= [\frac{1}{6} (x^{6} ln (x^{2}) - 6 \cdot \frac{x ^{6}}{18})]_{1}^{e}

Vereinfache

[\frac{1}{6} (x^{6} ln (x^{2}) - 6 \cdot \frac{x ^{6}}{18})]_{1}^{e} : [\frac{1}{6} (2 x^{6} ln (x) - \frac{x ^{6}}{3})]_{1}^{e}

= [\frac{1}{6} (2 x^{6} ln (x) - \frac{x ^{6}}{3})]_{1}^{e}

Berechne die Grenzen:

\frac{5 e ^{6}}{18} + \frac{1}{18}

= \frac{5 e ^{6}}{18} + \frac{1}{18}

Vereinfache

= \frac{5 e ^{6} + 1}{18}

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