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Beliebt Rechnen >

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

Lösung

\int (\frac{5 x ^{3} - 3 5 x}{5 x}) d x

Lösung

\frac{1}{3} (x^{3} - 9 5 x) + C

Schritte zur Lösung

\int \frac{5 x ^{3} - 3 5 x}{5 x} d x

Entferne die Konstante:

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

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

Multipliziere aus

\frac{5 x ^{3} - 3 5 x}{x} : 5 x^{2} - \frac{3}{x ^{\frac{4}{5}}}

= \frac{1}{5} \cdot \int 5 x^{2} - \frac{3}{x ^{\frac{4}{5}}} 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 5 x^{2} d x - \int \frac{3}{x ^{\frac{4}{5}}} d x)

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

\int \frac{3}{x ^{\frac{4}{5}}} d x = 15 x^{\frac{1}{5}}

= \frac{1}{5} (\frac{5 x ^{3}}{3} - 15 x^{\frac{1}{5}})

Vereinfache

\frac{1}{5} (\frac{5 x ^{3}}{3} - 15 x^{\frac{1}{5}}) : \frac{1}{3} (x^{3} - 9 5 x)

= \frac{1}{3} (x^{3} - 9 5 x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (x^{3} - 9 5 x) + C

Graph

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