integral of (6x^8+9x^2-9x)/(3x^2)

Lösung

\int \frac{6 x ^{8} + 9 x ^{2} - 9 x}{3 x ^{2}} d x

\frac{1}{7} (2 x^{7} + 21 x - 21 ln ∣ x ∣) + C

Schritte zur Lösung

\int \frac{6 x ^{8} + 9 x ^{2} - 9 x}{3 x ^{2}} 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{6 x ^{8} + 9 x ^{2} - 9 x}{x ^{2}} d x

Multipliziere aus

\frac{6 x ^{8} + 9 x ^{2} - 9 x}{x ^{2}} : 6 x^{6} + 9 - \frac{9}{x}

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

\int 6 x^{6} d x = \frac{6 x ^{7}}{7}

\int 9 d x = 9 x

\int \frac{9}{x} d x = 9 ln ∣ x ∣

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

Vereinfache

\frac{1}{3} (\frac{6 x ^{7}}{7} + 9 x - 9 ln ∣ x ∣) : \frac{1}{7} (2 x^{7} + 21 x - 21 ln ∣ x ∣)

= \frac{1}{7} (2 x^{7} + 21 x - 21 ln ∣ x ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{7} (2 x^{7} + 21 x - 21 ln ∣ x ∣) + C

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