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Beliebt Rechnen >

integral of (rsqrt(5-r^2))

Lösung

\int (r 5 - r^{2}) d r

Lösung

- \frac{1}{3} (5 - r^{2})^{\frac{3}{2}} + C

Schritte zur Lösung

\int r 5 - r^{2} d r

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 = 5 - r^{2}

ein

= - \frac{1}{2} \cdot \frac{2}{3} (5 - r^{2})^{\frac{3}{2}}

Vereinfache

- \frac{1}{2} \cdot \frac{2}{3} (5 - r^{2})^{\frac{3}{2}} : - \frac{1}{3} (5 - r^{2})^{\frac{3}{2}}

= - \frac{1}{3} (5 - r^{2})^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} (5 - r^{2})^{\frac{3}{2}} + C

Graph

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