integral of 3x(x^2-2x)

Lösung

\int 3 x (x^{2} - 2 x) d x

\frac{3 x ^{4}}{4} - 2 x^{3} + C

Schritte zur Lösung

\int 3 x (x^{2} - 2 x) d x

Entferne die Konstante:

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

= 3 \cdot \int x (x^{2} - 2 x) d x

Multipliziere aus

x (x^{2} - 2 x) : x^{3} - 2 x^{2}

= 3 \cdot \int x^{3} - 2 x^{2} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (\int x^{3} d x - \int 2 x^{2} d x)

\int x^{3} d x = \frac{x ^{4}}{4}

\int 2 x^{2} d x = \frac{2 x ^{3}}{3}

= 3 (\frac{x ^{4}}{4} - \frac{2 x ^{3}}{3})

Vereinfache

3 (\frac{x ^{4}}{4} - \frac{2 x ^{3}}{3}) : \frac{3 x ^{4}}{4} - 2 x^{3}

= \frac{3 x ^{4}}{4} - 2 x^{3}

Füge eine Konstante zur Lösung hinzu

= \frac{3 x ^{4}}{4} - 2 x^{3} + C

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