integral of (x^3+2x+5)/(x^2)

Lösung

\int \frac{x ^{3} + 2 x + 5}{x ^{2}} d x

\frac{x ^{2}}{2} + 2 ln ∣ x ∣ - \frac{5}{x} + C

Schritte zur Lösung

\int \frac{x ^{3} + 2 x + 5}{x ^{2}} d x

Multipliziere aus

\frac{x ^{3} + 2 x + 5}{x ^{2}} : x + \frac{2}{x} + \frac{5}{x ^{2}}

= \int x + \frac{2}{x} + \frac{5}{x ^{2}} 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 x d x + \int \frac{2}{x} d x + \int \frac{5}{x ^{2}} d x

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{5}{x ^{2}} d x = - \frac{5}{x}

= \frac{x ^{2}}{2} + 2 ln ∣ x ∣ - \frac{5}{x}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} + 2 ln ∣ x ∣ - \frac{5}{x} + C

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