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Beliebt Rechnen >

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

Lösung

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

Lösung

3 x - \frac{5}{3} ln ∣ x + 2 ∣ + \frac{23}{3} ln ∣ x - 4 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{3 x ^{2} - 2}{x ^{2} - 2 x - 8} : 3 - \frac{5}{3 ( x + 2 )} + \frac{23}{3 ( x - 4 )}

= \int 3 - \frac{5}{3 ( x + 2 )} + \frac{23}{3 ( x - 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 3 d x - \int \frac{5}{3 ( x + 2 )} d x + \int \frac{23}{3 ( x - 4 )} d x

\int 3 d x = 3 x

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

\int \frac{23}{3 ( x - 4 )} d x = \frac{23}{3} ln ∣ x - 4 ∣

= 3 x - \frac{5}{3} ln ∣ x + 2 ∣ + \frac{23}{3} ln ∣ x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= 3 x - \frac{5}{3} ln ∣ x + 2 ∣ + \frac{23}{3} ln ∣ x - 4 ∣ + C

Graph

Beliebte Beispiele

integral of (y-4)^2

\int (y - 4)^{2} d y

derivative of (x^2/(x-6))

\frac{d}{d x} (\frac{x ^{2}}{x - 6})

(\partial)/(\partial x)((xy)/(sqrt(1+xy^2)))

\frac{\partial}{\partial x} (\frac{x y}{1 + x y ^{2}})

derivative of ae^{2x}+be^x

\frac{d}{d x} (a e^{2 x} + b e^{x})

(\partial)/(\partial z)(e^xsin(yz))

\frac{\partial}{\partial z} (e^{x} sin (yz))