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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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