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Beliebt Rechnen >

integral of sqrt(1+4r^2)*r

Lösung

\int 1 + 4 r^{2} \cdot r d r

Lösung

\frac{1}{12} (1 + 4 r^{2})^{\frac{3}{2}} + C

Schritte zur Lösung

\int 1 + 4 r^{2} r d r

Wende U-Substitution an

= \int \frac{u}{8} d u

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int u d u

Wende die Potenzregel an

= \frac{1}{8} \cdot \frac{2}{3} u^{\frac{3}{2}}

Setze in

u = 1 + 4 r^{2}

ein

= \frac{1}{8} \cdot \frac{2}{3} (1 + 4 r^{2})^{\frac{3}{2}}

Vereinfache

\frac{1}{8} \cdot \frac{2}{3} (1 + 4 r^{2})^{\frac{3}{2}} : \frac{1}{12} (1 + 4 r^{2})^{\frac{3}{2}}

= \frac{1}{12} (1 + 4 r^{2})^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{12} (1 + 4 r^{2})^{\frac{3}{2}} + C

Graph

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