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Beliebt Rechnen >

integral of (-4x^2)/(x^4-1)

Lösung

\int \frac{- 4 x ^{2}}{x ^{4} - 1} d x

Lösung

- 2 arctan (x) + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ + C

Schritte zur Lösung

\int \frac{- 4 x ^{2}}{x ^{4} - 1} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 4 \cdot \int \frac{x ^{2}}{x ^{4} - 1} d x

Ermittle den Partialbruch von

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

= - 4 \cdot \int \frac{1}{2 ( x ^{2} + 1 )} - \frac{1}{4 ( x + 1 )} + \frac{1}{4 ( 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

= - 4 (\int \frac{1}{2 ( x ^{2} + 1 )} d x - \int \frac{1}{4 ( x + 1 )} d x + \int \frac{1}{4 ( x - 1 )} d x)

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

\int \frac{1}{4 ( x + 1 )} d x = \frac{1}{4} ln ∣ x + 1 ∣

\int \frac{1}{4 ( x - 1 )} d x = \frac{1}{4} ln ∣ x - 1 ∣

= - 4 (\frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣)

Vereinfache

- 4 (\frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣) : - 2 arctan (x) + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣

= - 2 arctan (x) + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= - 2 arctan (x) + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ + C

Graph

Beliebte Beispiele

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\int (e^{y} - y e^{- y}) d y