integral of (x^3-3x-8)/(x-4)

Lösung

\int \frac{x ^{3} - 3 x - 8}{x - 4} d x

\frac{x ^{3}}{3} + 2 x^{2} + 13 x + 44 ln ∣ x - 4 ∣ + 12 + C

Schritte zur Lösung

\int \frac{x ^{3} - 3 x - 8}{x - 4} d x

Multipliziere aus

\frac{x ^{3} - 3 x - 8}{x - 4} : \frac{x ^{3}}{x - 4} - \frac{3 x}{x - 4} - \frac{8}{x - 4}

Wende die Summenregel an:

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

= \int \frac{x ^{3}}{x - 4} d x - \int \frac{3 x}{x - 4} d x - \int \frac{8}{x - 4} d x

\int \frac{x ^{3}}{x - 4} d x = \frac{x ^{3}}{3} + 2 x^{2} + 16 x + 64 ln ∣ x - 4 ∣

\int \frac{3 x}{x - 4} d x = 3 (x - 4 + 4 ln ∣ x - 4 ∣)

\int \frac{8}{x - 4} d x = 8 ln ∣ x - 4 ∣

= \frac{x ^{3}}{3} + 2 x^{2} + 16 x + 64 ln ∣ x - 4 ∣ - 3 (x - 4 + 4 ln ∣ x - 4 ∣) - 8 ln ∣ x - 4 ∣

Vereinfache

\frac{x ^{3}}{3} + 2 x^{2} + 16 x + 64 ln ∣ x - 4 ∣ - 3 (x - 4 + 4 ln ∣ x - 4 ∣) - 8 ln ∣ x - 4 ∣ : \frac{x ^{3}}{3} + 2 x^{2} + 13 x + 44 ln ∣ x - 4 ∣ + 12

= \frac{x ^{3}}{3} + 2 x^{2} + 13 x + 44 ln ∣ x - 4 ∣ + 12

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + 2 x^{2} + 13 x + 44 ln ∣ x - 4 ∣ + 12 + C