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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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