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Beliebt Rechnen >

integral of (2x+6)/(x(x+1)^2)

Lösung

\int \frac{2 x + 6}{x ( x + 1 ) ^{2}} d x

Lösung

\frac{4}{x + 1} + 6 ln ∣ x ∣ - 6 ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int \frac{2 x + 6}{x ( x + 1 ) ^{2}} d x

Multipliziere aus

\frac{2 x + 6}{x ( x + 1 ) ^{2}} : \frac{2 x}{x ( x + 1 ) ^{2}} + \frac{6}{x ( 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{2 x}{x ( x + 1 ) ^{2}} d x + \int \frac{6}{x ( x + 1 ) ^{2}} d x

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

\int \frac{6}{x ( x + 1 ) ^{2}} d x = 6 (ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1})

= - \frac{2}{x + 1} + 6 (ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1})

Vereinfache

- \frac{2}{x + 1} + 6 (ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1}) : \frac{4}{x + 1} + 6 ln ∣ x ∣ - 6 ln ∣ x + 1 ∣

= \frac{4}{x + 1} + 6 ln ∣ x ∣ - 6 ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{4}{x + 1} + 6 ln ∣ x ∣ - 6 ln ∣ x + 1 ∣ + C

Graph

Beliebte Beispiele

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\int (5 x + 4) 5 1 + 8 x + 5 x^{2} d x

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\int e^{- x} arctan (e^{x}) d x

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\int z^{2} (z - 1)^{4} d z

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\int 3 x 25 - x^{2} d x

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