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

Lösung

\int (\frac{x ^{5} + x ^{2} - x + 5}{x}) d x

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

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{5} + x ^{2} - x + 5}{x} : x^{4} + x - 1 + \frac{5}{x}

= \int x^{4} + x - 1 + \frac{5}{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 x^{4} d x + \int x d x - \int 1 d x + \int \frac{5}{x} d x

\int x^{4} d x = \frac{x ^{5}}{5}

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

\int 1 d x = x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

Füge eine Konstante zur Lösung hinzu

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

s l o p e (2, - 4), (5, 8)

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