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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{( x ^{2} - 1 ) ( x ^{2} - 4 )} : \frac{1}{6 ( x + 1 )} - \frac{1}{6 ( x - 1 )} - \frac{1}{12 ( x + 2 )} + \frac{1}{12 ( x - 2 )}

= \int \frac{1}{6 ( x + 1 )} - \frac{1}{6 ( x - 1 )} - \frac{1}{12 ( x + 2 )} + \frac{1}{12 ( 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}{6 ( x + 1 )} d x - \int \frac{1}{6 ( x - 1 )} d x - \int \frac{1}{12 ( x + 2 )} d x + \int \frac{1}{12 ( x - 2 )} d x

\int \frac{1}{6 ( x + 1 )} d x = \frac{1}{6} ln ∣ x + 1 ∣

\int \frac{1}{6 ( x - 1 )} d x = \frac{1}{6} ln ∣ x - 1 ∣

\int \frac{1}{12 ( x + 2 )} d x = \frac{1}{12} ln ∣ x + 2 ∣

\int \frac{1}{12 ( x - 2 )} d x = \frac{1}{12} ln ∣ x - 2 ∣

= \frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

integral of (xe^x)/(sqrt(e^x+1))

\int \frac{x e ^{x}}{e ^{x} + 1} d x

integral of x^3sqrt(15+x^4)

\int x^{3} 15 + x^{4} d x

integral of 1/(\sqrt[4]{x^5)}

\int \frac{1}{4 x ^{5}} d x

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

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

integral of ln^2(y)

\int ln^{2} (y) d y