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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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