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Beliebt Rechnen >

integral of (x^4+\sqrt[3]{x}+3x)/(5x^3)

Lösung

Lösung

\frac{5 x ^{\frac{14}{3}} - 6 x - 30 x ^{\frac{5}{3}}}{50 x ^{\frac{8}{3}}} + C

Schritte zur Lösung

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Multipliziere aus

= \frac{1}{5} \cdot \int x + \frac{1}{x ^{\frac{8}{3}}} + \frac{3}{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

= \frac{1}{5} (\int x d x + \int \frac{1}{x ^{\frac{8}{3}}} d x + \int \frac{3}{x ^{2}} d x)

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

\int \frac{1}{x ^{\frac{8}{3}}} d x = - \frac{3}{5 x ^{\frac{5}{3}}}

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

= \frac{1}{5} (\frac{x ^{2}}{2} - \frac{3}{5 x ^{\frac{5}{3}}} - \frac{3}{x})

Vereinfache

\frac{1}{5} (\frac{x ^{2}}{2} - \frac{3}{5 x ^{\frac{5}{3}}} - \frac{3}{x}) : \frac{5 x ^{\frac{14}{3}} - 6 x - 30 x ^{\frac{5}{3}}}{50 x ^{\frac{8}{3}}}

= \frac{5 x ^{\frac{14}{3}} - 6 x - 30 x ^{\frac{5}{3}}}{50 x ^{\frac{8}{3}}}

Füge eine Konstante zur Lösung hinzu

= \frac{5 x ^{\frac{14}{3}} - 6 x - 30 x ^{\frac{5}{3}}}{50 x ^{\frac{8}{3}}} + C

Graph

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