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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{3} - 2 x ^{2} + x - 3}{x} : x^{2} - 2 x + 1 - \frac{3}{x}

= \int x^{2} - 2 x + 1 - \frac{3}{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 2 x d x + \int 1 d x - \int \frac{3}{x} d x

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

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

\int 1 d x = x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

Füge eine Konstante zur Lösung hinzu

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

\int (\frac{x}{2} + \frac{2}{x}) d x

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