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Beliebt Rechnen >

integral of (sqrt(x)+x^{1/3}+x)/(x^2)

Lösung

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

Lösung

- \frac{2}{x} - \frac{3}{2 x ^{\frac{2}{3}}} + ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x + x ^{\frac{1}{3}} + x}{x ^{2}} : \frac{1}{x ^{\frac{3}{2}}} + \frac{1}{x ^{\frac{5}{3}}} + \frac{1}{x}

= \int \frac{1}{x ^{\frac{3}{2}}} + \frac{1}{x ^{\frac{5}{3}}} + \frac{1}{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 ^{\frac{3}{2}}} d x + \int \frac{1}{x ^{\frac{5}{3}}} d x + \int \frac{1}{x} d x

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

\int \frac{1}{x ^{\frac{5}{3}}} d x = - \frac{3}{2 x ^{\frac{2}{3}}}

\int \frac{1}{x} d x = ln ∣ x ∣

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

Wende Radikal Regel an:

= - \frac{2}{x} - \frac{3}{2 x ^{\frac{2}{3}}} + ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{x} - \frac{3}{2 x ^{\frac{2}{3}}} + ln ∣ x ∣ + C

Graph

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