integral of x\sqrt[3]{(9+x^2)^2}

Lösung

\int x 3 (9 + x^{2})^{2} d x

\frac{3}{10} (9 + x^{2})^{\frac{5}{3}} + C

Schritte zur Lösung

\int x 3 (9 + x^{2})^{2} d x

Wende U-Substitution an

= \int \frac{( 9 + u ) ^{\frac{2}{3}}}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int (9 + u)^{\frac{2}{3}} d u

Wende U-Substitution an

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

Wende die Potenzregel an

= \frac{1}{2} \cdot \frac{3}{5} v^{\frac{5}{3}}

Ersetze zurück

= \frac{1}{2} \cdot \frac{3}{5} (9 + x^{2})^{\frac{5}{3}}

Vereinfache

\frac{1}{2} \cdot \frac{3}{5} (9 + x^{2})^{\frac{5}{3}} : \frac{3}{10} (9 + x^{2})^{\frac{5}{3}}

= \frac{3}{10} (9 + x^{2})^{\frac{5}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{10} (9 + x^{2})^{\frac{5}{3}} + C

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