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Beliebt Rechnen >

integral of (24x^3-48x^2+7)/(x^2-2x)

Lösung

\int \frac{24 x ^{3} - 48 x ^{2} + 7}{x ^{2} - 2 x} d x

Lösung

12 x^{2} - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

Schritte zur Lösung

\int \frac{24 x ^{3} - 48 x ^{2} + 7}{x ^{2} - 2 x} d x

Lange Division

\frac{24 x ^{3} - 48 x ^{2} + 7}{x ^{2} - 2 x} : 24 x + \frac{7}{x ^{2} - 2 x}

= \int 24 x + \frac{7}{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 24 x d x + \int \frac{7}{x ^{2} - 2 x} d x

\int 24 x d x = 12 x^{2}

\int \frac{7}{x ^{2} - 2 x} d x = - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

= 12 x^{2} - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Füge eine Konstante zur Lösung hinzu

= 12 x^{2} - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

Graph

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