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Beliebt Rechnen >

integral of (18x^3-36x^2+4)/(x^2-2x)

Lösung

\int \frac{18 x ^{3} - 36 x ^{2} + 4}{x ^{2} - 2 x} d x

Lösung

9 x^{2} - 4 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

Schritte zur Lösung

\int \frac{18 x ^{3} - 36 x ^{2} + 4}{x ^{2} - 2 x} d x

Lange Division

\frac{18 x ^{3} - 36 x ^{2} + 4}{x ^{2} - 2 x} : 18 x + \frac{4}{x ^{2} - 2 x}

= \int 18 x + \frac{4}{x ^{2} - 2 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 18 x d x + \int \frac{4}{x ^{2} - 2 x} d x

\int 18 x d x = 9 x^{2}

\int \frac{4}{x ^{2} - 2 x} d x = - 4 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

= 9 x^{2} - 4 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Füge eine Konstante zur Lösung hinzu

= 9 x^{2} - 4 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

Graph

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