integral from 1 to e of x^2ln(x^3)

Lösung

\int_{1}^{e} x^{2} ln (x^{3}) d x

\frac{2 e ^{3} + 1}{3}

Dezimale

13.72369 \dots

Schritte zur Lösung

\int_{1}^{e} x^{2} ln (x^{3}) d x

Wende die partielle Integration an

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

\int x^{2} d x = \frac{x ^{3}}{3}

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

Vereinfache

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

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

Berechne die Grenzen:

\frac{2 e ^{3}}{3} + \frac{1}{3}

= \frac{2 e ^{3}}{3} + \frac{1}{3}

Vereinfache

= \frac{2 e ^{3} + 1}{3}