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Beliebt Rechnen >

integral of (x^2-4x-4)/(x^3-2x^2+4x-8)

Lösung

\int \frac{x ^{2} - 4 x - 4}{x ^{3} - 2 x ^{2} + 4 x - 8} d x

Lösung

- ln ∣ x - 2 ∣ + ln x^{2} + 4 + C

Schritte zur Lösung

\int \frac{x ^{2} - 4 x - 4}{x ^{3} - 2 x ^{2} + 4 x - 8} d x

Ermittle den Partialbruch von

\frac{x ^{2} - 4 x - 4}{x ^{3} - 2 x ^{2} + 4 x - 8} : - \frac{1}{x - 2} + \frac{2 x}{x ^{2} + 4}

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

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

\int \frac{2 x}{x ^{2} + 4} d x = ln x^{2} + 4

= - ln ∣ x - 2 ∣ + ln x^{2} + 4

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x - 2 ∣ + ln x^{2} + 4 + C

Graph

Beliebte Beispiele

derivative f(x)=4x^3-1x-2

d er i v a t i v e f (x) = 4 x^{3} - 1 x - 2

(\partial)/(\partial y)(2xy)

\frac{\partial}{\partial y} (2 x y)

implicit (dy)/(dx),xy=24

im pl i c i t \frac{d y}{d x}, x y = 24

d/(d{r)}(4pi{r}^2)

\frac{d}{d r} (4 π r^{2})

integral of (x^2)/(sqrt(225-x^2))

\int \frac{x ^{2}}{225 - x ^{2}} d x