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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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\frac{d}{d t} (4 e^{t} \cdot t)

integral from 2 to 7 of x/(x^2+6x+13)

\int_{2}^{7} \frac{x}{x ^{2} + 6 x + 13} d x

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\frac{d}{d x} (e^{\frac{x}{4}})

integral of (sqrt(x^2-25))/(2x^2)

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

integral of x^2*sqrt(x^3+3)

\int x^{2} \cdot x^{3} + 3 d x