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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

\frac{1}{( u - 1 ) u ^{2}} : \frac{1}{u - 1} - \frac{1}{u} - \frac{1}{u ^{2}}

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

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}{u - 1} d u - \int \frac{1}{u} d u - \int \frac{1}{u ^{2}} d u

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u - 1 ∣ - ln ∣ u ∣ - (- \frac{1}{u})

Setze in

u = x + 1

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of xe^x+2e^x

\int x e^{x} + 2 e^{x} d x

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

\int \frac{2 x - 3}{x ^{3} - x ^{2}} d x

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

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

integral of 2e^{1/2 x}

\int 2 e^{\frac{1}{2} x} d x

integral of (2e^{-2x})/x

\int \frac{2 e ^{- 2 x}}{x} d x