integral of (x^2-1)ln(x)

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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