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Beliebt Rechnen >

integral of (2x^3+7x^2-6x+9)/(3x)

Lösung

\int \frac{2 x ^{3} + 7 x ^{2} - 6 x + 9}{3 x} d x

Lösung

\frac{1}{3} (\frac{2 x ^{3}}{3} + \frac{7 x ^{2}}{2} - 6 x + 9 ln ∣ x ∣) + C

Schritte zur Lösung

\int \frac{2 x ^{3} + 7 x ^{2} - 6 x + 9}{3 x} d x

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int \frac{2 x ^{3} + 7 x ^{2} - 6 x + 9}{x} d x

Multipliziere aus

\frac{2 x ^{3} + 7 x ^{2} - 6 x + 9}{x} : 2 x^{2} + 7 x - 6 + \frac{9}{x}

= \frac{1}{3} \cdot \int 2 x^{2} + 7 x - 6 + \frac{9}{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

= \frac{1}{3} (\int 2 x^{2} d x + \int 7 x d x - \int 6 d x + \int \frac{9}{x} d x)

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

\int 7 x d x = \frac{7 x ^{2}}{2}

\int 6 d x = 6 x

\int \frac{9}{x} d x = 9 ln ∣ x ∣

= \frac{1}{3} (\frac{2 x ^{3}}{3} + \frac{7 x ^{2}}{2} - 6 x + 9 ln ∣ x ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (\frac{2 x ^{3}}{3} + \frac{7 x ^{2}}{2} - 6 x + 9 ln ∣ x ∣) + C

Graph

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