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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{4} (1 - x^{\frac{3}{2}})^{\frac{8}{3}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

= - \frac{2}{3} \cdot \frac{3}{8} u^{\frac{8}{3}}

Setze in

u = 1 - x^{\frac{3}{2}}

ein

= - \frac{2}{3} \cdot \frac{3}{8} (1 - x^{\frac{3}{2}})^{\frac{8}{3}}

Vereinfache

- \frac{2}{3} \cdot \frac{3}{8} (1 - x^{\frac{3}{2}})^{\frac{8}{3}} : - \frac{1}{4} (1 - x^{\frac{3}{2}})^{\frac{8}{3}}

= - \frac{1}{4} (1 - x^{\frac{3}{2}})^{\frac{8}{3}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} (1 - x^{\frac{3}{2}})^{\frac{8}{3}} + C

Graph

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