integral of (x-x^3-2)sqrt(x)

Lösung

\int (x - x^{3} - 2) x d x

\frac{2}{5} x^{\frac{5}{2}} - \frac{2}{9} x^{\frac{9}{2}} - \frac{4}{3} x^{\frac{3}{2}} + C

Schritte zur Lösung

\int (x - x^{3} - 2) x d x

Multipliziere aus

(x - x^{3} - 2) x : x^{\frac{3}{2}} - x^{\frac{7}{2}} - 2 x

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

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

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

\int 2 x d x = \frac{4}{3} x^{\frac{3}{2}}

= \frac{2}{5} x^{\frac{5}{2}} - \frac{2}{9} x^{\frac{9}{2}} - \frac{4}{3} x^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} x^{\frac{5}{2}} - \frac{2}{9} x^{\frac{9}{2}} - \frac{4}{3} x^{\frac{3}{2}} + C

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