integral of x^2(3-x^3)^{2/3}

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

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

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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