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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 1 ∣ - \frac{3}{4} ln ∣ x - 1 ∣ - \frac{1}{2 ( x - 1 )} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \int \frac{1}{x} - \frac{1}{4 ( x + 1 )} - \frac{3}{4 ( x - 1 )} + \frac{1}{2 ( 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}{x} d x - \int \frac{1}{4 ( x + 1 )} d x - \int \frac{3}{4 ( x - 1 )} d x + \int \frac{1}{2 ( x - 1 ) ^{2}} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int \frac{3}{4 ( x - 1 )} d x = \frac{3}{4} ln ∣ x - 1 ∣

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

= ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 1 ∣ - \frac{3}{4} ln ∣ x - 1 ∣ - \frac{1}{2 ( x - 1 )}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 1 ∣ - \frac{3}{4} ln ∣ x - 1 ∣ - \frac{1}{2 ( x - 1 )} + C

Graph

Beliebte Beispiele

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\int 7^{3 x} d x

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\int (x + x)^{2} d x

limit as x approaches 2 of e^x-1

x \to 2 lim (e^{x} - 1)