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

Lösung

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

\frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{3}}{x ^{2} + x - 2} : x - 1 + \frac{1}{3 ( x - 1 )} + \frac{8}{3 ( x + 2 )}

= \int x - 1 + \frac{1}{3 ( x - 1 )} + \frac{8}{3 ( 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 x d x - \int 1 d x + \int \frac{1}{3 ( x - 1 )} d x + \int \frac{8}{3 ( x + 2 )} d x

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

\int 1 d x = x

\int \frac{1}{3 ( x - 1 )} d x = \frac{1}{3} ln ∣ x - 1 ∣

\int \frac{8}{3 ( x + 2 )} d x = \frac{8}{3} ln ∣ x + 2 ∣

= \frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣ + C

\int \frac{x ^{4}}{x ^{5} - 2} d x

\frac{d}{d x} (\frac{e ^{\frac{1}{x}}}{x ^{2}})

\frac{d}{d y} (- x y)

\frac{d y}{d x} = \frac{x ^{5}}{3 y ^{2}}

\frac{d}{d x} ((\frac{x}{9})^{2} + (\frac{y}{4})^{2}) = 1