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Beliebt Rechnen >

integral of (x-5)/(x^2(x+1))

Lösung

\int \frac{x - 5}{x ^{2} ( x + 1 )} d x

Lösung

6 ln ∣ x ∣ + \frac{5}{x} - 6 ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int \frac{x - 5}{x ^{2} ( x + 1 )} d x

Ermittle den Partialbruch von

\frac{x - 5}{x ^{2} ( x + 1 )} : \frac{6}{x} - \frac{5}{x ^{2}} - \frac{6}{x + 1}

= \int \frac{6}{x} - \frac{5}{x ^{2}} - \frac{6}{x + 1} 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{6}{x} d x - \int \frac{5}{x ^{2}} d x - \int \frac{6}{x + 1} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{5}{x ^{2}} d x = - \frac{5}{x}

\int \frac{6}{x + 1} d x = 6 ln ∣ x + 1 ∣

= 6 ln ∣ x ∣ - (- \frac{5}{x}) - 6 ln ∣ x + 1 ∣

Vereinfache

= 6 ln ∣ x ∣ + \frac{5}{x} - 6 ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= 6 ln ∣ x ∣ + \frac{5}{x} - 6 ln ∣ x + 1 ∣ + C

Graph

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