integral x^5(2-x^3)^{1/2}

Lösung

integral

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

- \frac{1}{3} (- \frac{2}{5} (2 - x^{3})^{\frac{5}{2}} + \frac{4}{3} (2 - x^{3})^{\frac{3}{2}}) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int - \frac{( - u + 2 ) u ^{\frac{1}{2}}}{3} d u

Entferne die Konstante:

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

= - \frac{1}{3} \cdot \int (- u + 2) u^{\frac{1}{2}} d u

Multipliziere aus

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

= - \frac{1}{3} \cdot \int - u^{\frac{3}{2}} + 2 u^{\frac{1}{2}} d u

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}{3} (- \int u^{\frac{3}{2}} d u + \int 2 u^{\frac{1}{2}} d u)

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

\int 2 u^{\frac{1}{2}} d u = \frac{4}{3} u^{\frac{3}{2}}

= - \frac{1}{3} (- \frac{2}{5} u^{\frac{5}{2}} + \frac{4}{3} u^{\frac{3}{2}})

Setze in

u = 2 - x^{3}

ein

= - \frac{1}{3} (- \frac{2}{5} (2 - x^{3})^{\frac{5}{2}} + \frac{4}{3} (2 - x^{3})^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} (- \frac{2}{5} (2 - x^{3})^{\frac{5}{2}} + \frac{4}{3} (2 - x^{3})^{\frac{3}{2}}) + C

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