integral of (x^3-5x^2+3x+14)/(x-3)

Lösung

\int \frac{x ^{3} - 5 x ^{2} + 3 x + 14}{x - 3} d x

\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - 18 x + 5 ln ∣ x - 3 ∣ - \frac{5}{2} (x - 3)^{2} + 81 + C

Schritte zur Lösung

\int \frac{x ^{3} - 5 x ^{2} + 3 x + 14}{x - 3} d x

Multipliziere aus

\frac{x ^{3} - 5 x ^{2} + 3 x + 14}{x - 3} : \frac{x ^{3}}{x - 3} - \frac{5 x ^{2}}{x - 3} + \frac{3 x}{x - 3} + \frac{14}{x - 3}

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 - 3} d x - \int \frac{5 x ^{2}}{x - 3} d x + \int \frac{3 x}{x - 3} d x + \int \frac{14}{x - 3} d x

\int \frac{x ^{3}}{x - 3} d x = \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 9 x + 27 ln ∣ x - 3 ∣

\int \frac{5 x ^{2}}{x - 3} d x = 5 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣)

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

\int \frac{14}{x - 3} d x = 14 ln ∣ x - 3 ∣

= \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 9 x + 27 ln ∣ x - 3 ∣ - 5 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣) + 3 (x - 3 + 3 ln ∣ x - 3 ∣) + 14 ln ∣ x - 3 ∣

Vereinfache

\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 9 x + 27 ln ∣ x - 3 ∣ - 5 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣) + 3 (x - 3 + 3 ln ∣ x - 3 ∣) + 14 ln ∣ x - 3 ∣ : \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - 18 x + 5 ln ∣ x - 3 ∣ - \frac{5}{2} (x - 3)^{2} + 81

= \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - 18 x + 5 ln ∣ x - 3 ∣ - \frac{5}{2} (x - 3)^{2} + 81

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - 18 x + 5 ln ∣ x - 3 ∣ - \frac{5}{2} (x - 3)^{2} + 81 + C

t an g e n t f (x) = e^{- 3 x}, (0, 1)

\frac{d x}{d t} = 5 x (6 - x)

\frac{d y}{d x} + 3 y = 50

\frac{d y}{d x} = \frac{y}{x} + \frac{x}{y ^{5}}

\int \frac{12 u}{u ( u - 8 )} d u