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Beliebt Rechnen >

integral of (x-1)/(x^3+x^2-4x-4)

Lösung

\int \frac{x - 1}{x ^{3} + x ^{2} - 4 x - 4} d x

Lösung

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

Schritte zur Lösung

\int \frac{x - 1}{x ^{3} + x ^{2} - 4 x - 4} d x

Ermittle den Partialbruch von

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

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

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

\int \frac{3}{4 ( x + 2 )} d x = \frac{3}{4} ln ∣ x + 2 ∣

\int \frac{1}{12 ( x - 2 )} d x = \frac{1}{12} ln ∣ x - 2 ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (sec^2(2x))

\int (sec^{2} (2 x)) d x

integral of xsqrt(4x^2-1)

\int x 4 x^{2} - 1 d x

integral of e^{x/(20)}

\int e^{\frac{x}{20}} d x

integral of sqrt(x)(9x^3-2sqrt(x)+8/x)

\int x (9 x^{3} - 2 x + \frac{8}{x}) d x

integral of x(e^x-1)

\int x (e^{x} - 1) d x