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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Faktorisiere

\frac{x ^{3} + 2}{x ^{3} + x ^{2}}

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

Ermittle den Partialbruch von

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

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

\int 1 d x = x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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