integral from 0 to k of x(2x)/(k^2)

Lösung

\int_{0}^{k} x \frac{2 x}{k ^{2}} d x

\frac{2 k}{3}

Schritte zur Lösung

\int_{0}^{k} x \frac{2 x}{k ^{2}} d x

Entferne die Konstante:

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

= \frac{2}{k ^{2}} \cdot \int_{0}^{k} xx d x

Vereinfache

xx : x^{2}

= \frac{2}{k ^{2}} \cdot \int_{0}^{k} x^{2} d x

Wende die Potenzregel an

= \frac{2}{k ^{2}} [\frac{x ^{3}}{3}]_{0}^{k}

Berechne die Grenzen:

\frac{k ^{3}}{3}

= \frac{2}{k ^{2}} \cdot \frac{k ^{3}}{3}

Vereinfache

= \frac{2 k}{3}

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