integral of (3x^2-2)/(x^2-4x-12)

Lösung

\int \frac{3 x ^{2} - 2}{x ^{2} - 4 x - 12} d x

3 x - \frac{5}{4} ln ∣ x + 2 ∣ + \frac{53}{4} ln ∣ x - 6 ∣ + C

Schritte zur Lösung

\int \frac{3 x ^{2} - 2}{x ^{2} - 4 x - 12} d x

Ermittle den Partialbruch von

\frac{3 x ^{2} - 2}{x ^{2} - 4 x - 12} : 3 - \frac{5}{4 ( x + 2 )} + \frac{53}{4 ( x - 6 )}

= \int 3 - \frac{5}{4 ( x + 2 )} + \frac{53}{4 ( x - 6 )} 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}{4 ( x + 2 )} d x + \int \frac{53}{4 ( x - 6 )} d x

\int 3 d x = 3 x

\int \frac{5}{4 ( x + 2 )} d x = \frac{5}{4} ln ∣ x + 2 ∣

\int \frac{53}{4 ( x - 6 )} d x = \frac{53}{4} ln ∣ x - 6 ∣

= 3 x - \frac{5}{4} ln ∣ x + 2 ∣ + \frac{53}{4} ln ∣ x - 6 ∣

Füge eine Konstante zur Lösung hinzu

= 3 x - \frac{5}{4} ln ∣ x + 2 ∣ + \frac{53}{4} ln ∣ x - 6 ∣ + C

s l o p e y = x

\frac{\partial}{\partial w} (x e^{w})

\int_{0}^{6} x 36 - x^{2} d x

\int cos (1) d x

\frac{d ^{3}}{d x ^{3}} (x e^{x})