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Beliebt Rechnen >

integral of (x-9)/(x^3-5x^2+6x)

Lösung

\int \frac{x - 9}{x ^{3} - 5 x ^{2} + 6 x} d x

Lösung

- \frac{3}{2} ln ∣ x ∣ + \frac{7}{2} ln ∣ x - 2 ∣ - 2 ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{x - 9}{x ^{3} - 5 x ^{2} + 6 x} d x

Ermittle den Partialbruch von

\frac{x - 9}{x ^{3} - 5 x ^{2} + 6 x} : - \frac{3}{2 x} + \frac{7}{2 ( x - 2 )} - \frac{2}{x - 3}

= \int - \frac{3}{2 x} + \frac{7}{2 ( x - 2 )} - \frac{2}{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

= - \int \frac{3}{2 x} d x + \int \frac{7}{2 ( x - 2 )} d x - \int \frac{2}{x - 3} d x

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

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

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

= - \frac{3}{2} ln ∣ x ∣ + \frac{7}{2} ln ∣ x - 2 ∣ - 2 ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{2} ln ∣ x ∣ + \frac{7}{2} ln ∣ x - 2 ∣ - 2 ln ∣ x - 3 ∣ + C

Graph

Beliebte Beispiele

integral of (sin(x))/(1+tan^2(x))

\int \frac{sin ( x )}{1 + tan ^{2} ( x )} d x

integral of 2/7 (3^x)

\int \frac{2}{7} (3^{x}) d x

integral of ln(e^{4x})

\int ln (e^{4 x}) d x

integral of-2-2tan^2(x)

\int - 2 - 2 tan^{2} (x) d x

integral of (x^2-1)/(1+x^2)

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