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Beliebt Rechnen >

integral of (-10x^2)/((x+1)^3)

Lösung

\int \frac{- 10 x ^{2}}{( x + 1 ) ^{3}} d x

Lösung

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

Schritte zur Lösung

\int \frac{- 10 x ^{2}}{( x + 1 ) ^{3}} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 10 \cdot \int \frac{x ^{2}}{( x + 1 ) ^{3}} d x

Wende die partielle Integration an

= - 10 (\frac{x ^{3}}{( x + 1 ) ^{3}} - \int \frac{x ^{2} ( - x + 2 )}{( x + 1 ) ^{4}} d x)

\int \frac{x ^{2} ( - x + 2 )}{( x + 1 ) ^{4}} d x = - ln ∣ x + 1 ∣ - \frac{3}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - \frac{1}{3 ( x + 1 ) ^{3}} - \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} + \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}

= - 10 (\frac{x ^{3}}{( x + 1 ) ^{3}} - (- ln ∣ x + 1 ∣ - \frac{3}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - \frac{1}{3 ( x + 1 ) ^{3}} - \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} + \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}))

Vereinfache

= - 10 (\frac{x ^{3}}{( x + 1 ) ^{3}} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}})

Füge eine Konstante zur Lösung hinzu

= - 10 (\frac{x ^{3}}{( x + 1 ) ^{3}} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}) + C

Graph

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