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Beliebt Rechnen >

integral of (x^2+4x-1)/(x^3-x)

Lösung

\int \frac{x ^{2} + 4 x - 1}{x ^{3} - x} d x

Lösung

ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + 2 ln ∣ x - 1 ∣ + C

Schritte zur Lösung

\int \frac{x ^{2} + 4 x - 1}{x ^{3} - x} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 4 x - 1}{x ^{3} - x} : \frac{1}{x} - \frac{2}{x + 1} + \frac{2}{x - 1}

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{2}{x + 1} d x = 2 ln ∣ x + 1 ∣

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

= ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + 2 ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + 2 ln ∣ x - 1 ∣ + C

Graph

Beliebte Beispiele

f(x)=arctan(sqrt((1-x)/(1+x)))

f (x) = arctan (\frac{1 - x}{1 + x})

derivative of e^{(x+ln(x)})

\frac{d}{d x} (e^{(x + l n (x))})

fläche y=sin(x),y=sin(2x),x=0,x= pi/3

a re a y = sin (x), y = sin (2 x), x = 0, x = \frac{π}{3}

derivative of 1-ln(cos(x))

\frac{d}{d x} (1 - ln (cos (x)))

derivative f(x)=(x^2-4)/(x-3)

d er i v a t i v e f (x) = \frac{x ^{2} - 4}{x - 3}