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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= ln ∣ x ∣ - \frac{3}{x} - ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

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

\int (\frac{x ^{2}}{x + 3}) d x

\int 2 e^{\frac{1}{2} x} d x

\int \frac{2 e ^{- 2 x}}{x} d x

\int \frac{1}{xx} d x

\int sin (x) csc^{2} (x) d x