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Beliebt Rechnen >

integral 1/((x^2-1)^3)

Lösung

integral

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

Lösung

- \frac{3}{16} ln ∣ x + 1 ∣ + \frac{3}{16 ( x + 1 )} + \frac{1}{16 ( x + 1 ) ^{2}} + \frac{3}{16} ln ∣ x - 1 ∣ + \frac{3}{16 ( x - 1 )} - \frac{1}{16 ( x - 1 ) ^{2}} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{( x ^{2} - 1 ) ^{3}} : - \frac{3}{16 ( x + 1 )} - \frac{3}{16 ( x + 1 ) ^{2}} - \frac{1}{8 ( x + 1 ) ^{3}} + \frac{3}{16 ( x - 1 )} - \frac{3}{16 ( x - 1 ) ^{2}} + \frac{1}{8 ( x - 1 ) ^{3}}

= \int - \frac{3}{16 ( x + 1 )} - \frac{3}{16 ( x + 1 ) ^{2}} - \frac{1}{8 ( x + 1 ) ^{3}} + \frac{3}{16 ( x - 1 )} - \frac{3}{16 ( x - 1 ) ^{2}} + \frac{1}{8 ( x - 1 ) ^{3}} 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{3}{16 ( x + 1 )} d x - \int \frac{3}{16 ( x + 1 ) ^{2}} d x - \int \frac{1}{8 ( x + 1 ) ^{3}} d x + \int \frac{3}{16 ( x - 1 )} d x - \int \frac{3}{16 ( x - 1 ) ^{2}} d x + \int \frac{1}{8 ( x - 1 ) ^{3}} d x

\int \frac{3}{16 ( x + 1 )} d x = \frac{3}{16} ln ∣ x + 1 ∣

\int \frac{3}{16 ( x + 1 ) ^{2}} d x = - \frac{3}{16 ( x + 1 )}

\int \frac{1}{8 ( x + 1 ) ^{3}} d x = - \frac{1}{16 ( x + 1 ) ^{2}}

\int \frac{3}{16 ( x - 1 )} d x = \frac{3}{16} ln ∣ x - 1 ∣

\int \frac{3}{16 ( x - 1 ) ^{2}} d x = - \frac{3}{16 ( x - 1 )}

\int \frac{1}{8 ( x - 1 ) ^{3}} d x = - \frac{1}{16 ( x - 1 ) ^{2}}

= - \frac{3}{16} ln ∣ x + 1 ∣ - (- \frac{3}{16 ( x + 1 )}) - (- \frac{1}{16 ( x + 1 ) ^{2}}) + \frac{3}{16} ln ∣ x - 1 ∣ - (- \frac{3}{16 ( x - 1 )}) - \frac{1}{16 ( x - 1 ) ^{2}}

Vereinfache

= - \frac{3}{16} ln ∣ x + 1 ∣ + \frac{3}{16 ( x + 1 )} + \frac{1}{16 ( x + 1 ) ^{2}} + \frac{3}{16} ln ∣ x - 1 ∣ + \frac{3}{16 ( x - 1 )} - \frac{1}{16 ( x - 1 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{16} ln ∣ x + 1 ∣ + \frac{3}{16 ( x + 1 )} + \frac{1}{16 ( x + 1 ) ^{2}} + \frac{3}{16} ln ∣ x - 1 ∣ + \frac{3}{16 ( x - 1 )} - \frac{1}{16 ( x - 1 ) ^{2}} + C

Graph

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