integral of (x-x^3)/(sqrt(x))

Lösung

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

\frac{2}{3} x^{\frac{3}{2}} - \frac{2}{7} x^{\frac{7}{2}} + C

Schritte zur Lösung

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

Vereinfache

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

= \int x (1 - x^{2}) d x

Multipliziere aus

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

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

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

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

= \frac{2}{3} x^{\frac{3}{2}} - \frac{2}{7} x^{\frac{7}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} x^{\frac{3}{2}} - \frac{2}{7} x^{\frac{7}{2}} + C

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