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Beliebt Rechnen >

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

Lösung

\int \frac{1}{x ( 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

derivative of e^{-2x}*cos(3x)

\frac{d}{d x} (e^{- 2 x} \cdot cos (3 x))

limit as x approaches 0+of 3+x^5ln(x)

x \to 0 + lim (3 + x^{5} ln (x))

(\partial)/(\partial y)(x^3+yz^2)

\frac{\partial}{\partial y} (x^{3} + y z^{2})

integral of 1/(sqrt(x-2))

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

derivative of csc(cot(x))

\frac{d}{d x} (csc (cot (x)))