integral of 4x(x-2)^3

Lösung

\int 4 x (x - 2)^{3} d x

4 (\frac{x ^{5}}{5} - \frac{3 x ^{4}}{2} + 4 x^{3} - 4 x^{2}) + C

Schritte zur Lösung

\int 4 x (x - 2)^{3} d x

Entferne die Konstante:

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

= 4 \cdot \int x (x - 2)^{3} d x

Multipliziere aus

x (x - 2)^{3} : x^{4} - 6 x^{3} + 12 x^{2} - 8 x

= 4 \cdot \int x^{4} - 6 x^{3} + 12 x^{2} - 8 x 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

= 4 (\int x^{4} d x - \int 6 x^{3} d x + \int 12 x^{2} d x - \int 8 x d x)

\int x^{4} d x = \frac{x ^{5}}{5}

\int 6 x^{3} d x = \frac{3 x ^{4}}{2}

\int 12 x^{2} d x = 4 x^{3}

\int 8 x d x = 4 x^{2}

= 4 (\frac{x ^{5}}{5} - \frac{3 x ^{4}}{2} + 4 x^{3} - 4 x^{2})

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{x ^{5}}{5} - \frac{3 x ^{4}}{2} + 4 x^{3} - 4 x^{2}) + C

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