integral of (x^2)/2 ln(x)

Lösung

\int \frac{x ^{2}}{2} ln (x) d x

\frac{1}{2} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) + C

Schritte zur Lösung

\int \frac{x ^{2}}{2} ln (x) d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Wende die partielle Integration an

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

\int \frac{x ^{2}}{3} d x = \frac{x ^{3}}{9}

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) + C

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