integral of x(sqrt(r^2-x^2))

Lösung

\int x (r^{2} - x^{2}) d x

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

Schritte zur Lösung

\int x r^{2} - x^{2} d x

Wende U-Substitution an

= \int - \frac{u}{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 u d u

Wende die Potenzregel an

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

Setze in

u = r^{2} - x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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