integral of (x^3-x^2-12)/x

Lösung

\int \frac{x ^{3} - x ^{2} - 12}{x} d x

\frac{x ^{3}}{3} - \frac{x ^{2}}{2} - 12 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{x ^{3} - x ^{2} - 12}{x} d x

Multipliziere aus

\frac{x ^{3} - x ^{2} - 12}{x} : x^{2} - x - \frac{12}{x}

= \int x^{2} - x - \frac{12}{x} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} d x - \int x d x - \int \frac{12}{x} d x

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

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

\int \frac{12}{x} d x = 12 ln ∣ x ∣

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} - 12 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} - 12 ln ∣ x ∣ + C

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