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Beliebt Rechnen >

integral of x^{-1}(x^{-2}+x^{-1}+2+x^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

x^{- 1} (x^{- 2} + x^{- 1} + 2 + x^{2}) : \frac{1 + x + 2 x ^{2} + x ^{4}}{x ^{3}}

= \int \frac{1 + x + 2 x ^{2} + x ^{4}}{x ^{3}} d x

Multipliziere aus

\frac{1 + x + 2 x ^{2} + x ^{4}}{x ^{3}} : \frac{1}{x ^{3}} + \frac{1}{x ^{2}} + \frac{2}{x} + x

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

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int x d x = \frac{x ^{2}}{2}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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