integral from 0 to x of 20(3/2 x)^{1/3}

Lösung

\int_{0}^{x} 20 (\frac{3}{2} x)^{\frac{1}{3}} d x

\frac{15 \cdot 1 2 ^{\frac{1}{3}} x ^{\frac{4}{3}}}{2}

Schritte zur Lösung

\int_{0}^{x} 20 (\frac{3}{2} x)^{\frac{1}{3}} d x

Entferne die Konstante:

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

= 20 \cdot \int_{0}^{x} (\frac{3}{2} x)^{\frac{1}{3}} d x

Vereinfache

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

= 20 \cdot \int_{0}^{x} (\frac{3}{2})^{\frac{1}{3}} x^{\frac{1}{3}} d x

Entferne die Konstante:

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

= 20 (\frac{3}{2})^{\frac{1}{3}} \cdot \int_{0}^{x} x^{\frac{1}{3}} d x

Wende die Potenzregel an

= 20 (\frac{3}{2})^{\frac{1}{3}} [\frac{3}{4} x^{\frac{4}{3}}]_{0}^{x}

Vereinfache

20 (\frac{3}{2})^{\frac{1}{3}} [\frac{3}{4} x^{\frac{4}{3}}]_{0}^{x} : 10 \cdot 1 2^{\frac{1}{3}} [\frac{3}{4} x^{\frac{4}{3}}]_{0}^{x}

= 10 \cdot 1 2^{\frac{1}{3}} [\frac{3}{4} x^{\frac{4}{3}}]_{0}^{x}

Berechne die Grenzen:

\frac{3}{4} x^{\frac{4}{3}}

= 10 \cdot 1 2^{\frac{1}{3}} \cdot \frac{3}{4} x^{\frac{4}{3}}

Vereinfache

= \frac{15 \cdot 1 2 ^{\frac{1}{3}} x ^{\frac{4}{3}}}{2}

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