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Beliebt Rechnen >

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

Lösung

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

Lösung

x + ln ∣ x ∣ + \frac{2}{x} - 5 ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{3} - 2 x ^{2} - 4}{x ^{3} + 2 x ^{2}} : 1 + \frac{1}{x} - \frac{2}{x ^{2}} - \frac{5}{x + 2}

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int \frac{5}{x + 2} d x = 5 ln ∣ x + 2 ∣

= x + ln ∣ x ∣ - (- \frac{2}{x}) - 5 ln ∣ x + 2 ∣

Vereinfache

= x + ln ∣ x ∣ + \frac{2}{x} - 5 ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= x + ln ∣ x ∣ + \frac{2}{x} - 5 ln ∣ x + 2 ∣ + C

Graph

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\int \frac{e ^{x}}{x ^{3}} d x

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\int \frac{7}{16 + x ^{2}} d x

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\int x^{2} (x^{3} + 4)^{10} d x