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Beliebt Rechnen >

integral of (x-1)/(x^3-x^2-2x)

Lösung

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

Lösung

\frac{1}{2} ln ∣ x ∣ - \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{6} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

\int \frac{2}{3 ( x + 1 )} d x = \frac{2}{3} ln ∣ x + 1 ∣

\int \frac{1}{6 ( x - 2 )} d x = \frac{1}{6} ln ∣ x - 2 ∣

= \frac{1}{2} ln ∣ x ∣ - \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{6} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x ∣ - \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{6} ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

integral of 5sin^3(xco)s^5x

\int 5 sin^{3} (x co) s^{5} x d x

derivative ce^{2x}

d er i v a t i v e c e^{2 x}

fläche 8x^2,sqrt(1/8 x)

a re a 8 x^{2}, \frac{1}{8} x

(\partial)/(\partial x)(3x^4y^5+5x^7y^8)

\frac{\partial}{\partial x} (3 x^{4} y^{5} + 5 x^{7} y^{8})

integral from 0 to 3 of 1/(x-1)

\int_{0}^{3} \frac{1}{x - 1} d x