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Beliebt Rechnen >

integral of 3x(3x-1)^2

Lösung

\int 3 x (3 x - 1)^{2} d x

Lösung

3 (\frac{9 x ^{4}}{4} - 2 x^{3} + \frac{x ^{2}}{2}) + C

Schritte zur Lösung

\int 3 x (3 x - 1)^{2} d x

Entferne die Konstante:

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

= 3 \cdot \int x (3 x - 1)^{2} d x

Multipliziere aus

x (3 x - 1)^{2} : 9 x^{3} - 6 x^{2} + x

= 3 \cdot \int 9 x^{3} - 6 x^{2} + 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

= 3 (\int 9 x^{3} d x - \int 6 x^{2} d x + \int x d x)

\int 9 x^{3} d x = \frac{9 x ^{4}}{4}

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

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

= 3 (\frac{9 x ^{4}}{4} - 2 x^{3} + \frac{x ^{2}}{2})

Füge eine Konstante zur Lösung hinzu

= 3 (\frac{9 x ^{4}}{4} - 2 x^{3} + \frac{x ^{2}}{2}) + C

Graph

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