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Beliebt Rechnen >

integral of (2x-1)/(x^3-6x^2+11x-6)

Lösung

\int \frac{2 x - 1}{x ^{3} - 6 x ^{2} + 11 x - 6} d x

Lösung

\frac{1}{2} ln ∣ x - 1 ∣ - 3 ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{2 x - 1}{x ^{3} - 6 x ^{2} + 11 x - 6} d x

Ermittle den Partialbruch von

\frac{2 x - 1}{x ^{3} - 6 x ^{2} + 11 x - 6} : \frac{1}{2 ( x - 1 )} - \frac{3}{x - 2} + \frac{5}{2 ( x - 3 )}

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

\int \frac{1}{2 ( x - 1 )} d x = \frac{1}{2} ln ∣ x - 1 ∣

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

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

= \frac{1}{2} ln ∣ x - 1 ∣ - 3 ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x - 1 ∣ - 3 ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 3 ∣ + C

Graph

Beliebte Beispiele

limit as x approaches-infinity of-6/x

x \to - \infty lim (- \frac{6}{x})

y^'+2y=e^{(4t)}

y^{^{'}} + 2 y = e^{(4 t)}

integral of (x^2-x)

\int (x^{2} - x) d x

derivative of-1.8x^2+14.4x

\frac{d}{d x} (- 1.8 x^{2} + 14.4 x)

(\partial)/(\partial x)(-4(x+y^6))

\frac{\partial}{\partial x} (- 4 (x + y^{6}))