integral of (x-3)/(x^4-3x^2+2x)

Lösung

\int \frac{x - 3}{x ^{4} - 3 x ^{2} + 2 x} d x

- \frac{3}{2} ln ∣ x ∣ + \frac{11}{9} ln ∣ x - 1 ∣ + \frac{2}{3 ( x - 1 )} + \frac{5}{18} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

\int \frac{x - 3}{x ^{4} - 3 x ^{2} + 2 x} d x

Ermittle den Partialbruch von

\frac{x - 3}{x ^{4} - 3 x ^{2} + 2 x} : - \frac{3}{2 x} + \frac{11}{9 ( x - 1 )} - \frac{2}{3 ( x - 1 ) ^{2}} + \frac{5}{18 ( x + 2 )}

= \int - \frac{3}{2 x} + \frac{11}{9 ( x - 1 )} - \frac{2}{3 ( x - 1 ) ^{2}} + \frac{5}{18 ( x + 2 )} 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

= - \int \frac{3}{2 x} d x + \int \frac{11}{9 ( x - 1 )} d x - \int \frac{2}{3 ( x - 1 ) ^{2}} d x + \int \frac{5}{18 ( x + 2 )} d x

\int \frac{3}{2 x} d x = \frac{3}{2} ln ∣ x ∣

\int \frac{11}{9 ( x - 1 )} d x = \frac{11}{9} ln ∣ x - 1 ∣

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

\int \frac{5}{18 ( x + 2 )} d x = \frac{5}{18} ln ∣ x + 2 ∣

= - \frac{3}{2} ln ∣ x ∣ + \frac{11}{9} ln ∣ x - 1 ∣ - (- \frac{2}{3 ( x - 1 )}) + \frac{5}{18} ln ∣ x + 2 ∣

Vereinfache

= - \frac{3}{2} ln ∣ x ∣ + \frac{11}{9} ln ∣ x - 1 ∣ + \frac{2}{3 ( x - 1 )} + \frac{5}{18} ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{2} ln ∣ x ∣ + \frac{11}{9} ln ∣ x - 1 ∣ + \frac{2}{3 ( x - 1 )} + \frac{5}{18} ln ∣ x + 2 ∣ + C

d er i v a t i v e ln (7 - 6 x)

s l o p e (32, - 39), (17, 6)

\frac{d}{d x} (\frac{x ^{12}}{x ^{- 2}})

\frac{\partial}{\partial x} (\frac{x ^{2} + 1}{x ^{2} - 1})

x \to 0 - lim (2 x e^{2 x} + k x)