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Beliebt Rechnen >

integral of (6x)/(x^3-4x^2+x+6)

Lösung

\int \frac{6 x}{x ^{3} - 4 x ^{2} + x + 6} d x

Lösung

6 (- \frac{1}{12} ln ∣ x + 1 ∣ - \frac{2}{3} ln ∣ x - 2 ∣ + \frac{3}{4} ln ∣ x - 3 ∣) + C

Schritte zur Lösung

\int \frac{6 x}{x ^{3} - 4 x ^{2} + x + 6} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int \frac{x}{x ^{3} - 4 x ^{2} + x + 6} d x

Ermittle den Partialbruch von

\frac{x}{x ^{3} - 4 x ^{2} + x + 6} : - \frac{1}{12 ( x + 1 )} - \frac{2}{3 ( x - 2 )} + \frac{3}{4 ( x - 3 )}

= 6 \cdot \int - \frac{1}{12 ( x + 1 )} - \frac{2}{3 ( x - 2 )} + \frac{3}{4 ( x - 3 )} 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

= 6 (- \int \frac{1}{12 ( x + 1 )} d x - \int \frac{2}{3 ( x - 2 )} d x + \int \frac{3}{4 ( x - 3 )} d x)

\int \frac{1}{12 ( x + 1 )} d x = \frac{1}{12} ln ∣ x + 1 ∣

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

\int \frac{3}{4 ( x - 3 )} d x = \frac{3}{4} ln ∣ x - 3 ∣

= 6 (- \frac{1}{12} ln ∣ x + 1 ∣ - \frac{2}{3} ln ∣ x - 2 ∣ + \frac{3}{4} ln ∣ x - 3 ∣)

Füge eine Konstante zur Lösung hinzu

= 6 (- \frac{1}{12} ln ∣ x + 1 ∣ - \frac{2}{3} ln ∣ x - 2 ∣ + \frac{3}{4} ln ∣ x - 3 ∣) + C

Graph

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