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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= - ln ∣ x ∣ - (- \frac{1}{x}) + ln ∣ x - 2 ∣

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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