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

Lösung

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

- \frac{2}{9} (2 - x^{3})^{\frac{3}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende U-Substitution an

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

Wende die Potenzregel an

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

Ersetze zurück

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

Vereinfache

\frac{1}{3} (- \frac{2}{3} (2 - x^{3})^{\frac{3}{2}}) : - \frac{2}{9} (2 - x^{3})^{\frac{3}{2}}

= - \frac{2}{9} (2 - x^{3})^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{9} (2 - x^{3})^{\frac{3}{2}} + C

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