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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{x} + 3 ln ∣ x - 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

= - (- \frac{1}{x}) + 3 ln ∣ x - 1 ∣

Vereinfache

= \frac{1}{x} + 3 ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{x} + 3 ln ∣ x - 1 ∣ + C

Graph

Beliebte Beispiele

y^{''}+49y=cos(10t)

y^{^{''}} + 49 y = cos (10 t)

derivative ln(x^3+5x+3)

d er i v a t i v e ln (x^{3} + 5 x + 3)

limit as x approaches 1 of xsin(1/x)

x \to 1 lim (x sin (\frac{1}{x}))

tangent y=x^4-3x^2+2,(1,0)

t an g e n t y = x^{4} - 3 x^{2} + 2, (1, 0)

limit as x approaches 2-of (|x-x|)/(x-2)

x \to 2 - lim (\frac{∣ x - x ∣}{x - 2})