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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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