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Beliebt Rechnen >

integral of ((x^4-1))/(x^2(x+1)^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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