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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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