integral of ((x-9))/(x^2-x-2)

Lösung

\int \frac{( x - 9 )}{x ^{2} - x - 2} d x

\frac{10}{3} ln ∣ x + 1 ∣ - \frac{7}{3} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{x - 9}{x ^{2} - x - 2} d x

Ermittle den Partialbruch von

\frac{x - 9}{x ^{2} - x - 2} : \frac{10}{3 ( x + 1 )} - \frac{7}{3 ( x - 2 )}

= \int \frac{10}{3 ( x + 1 )} - \frac{7}{3 ( x - 2 )} 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{10}{3 ( x + 1 )} d x - \int \frac{7}{3 ( x - 2 )} d x

\int \frac{10}{3 ( x + 1 )} d x = \frac{10}{3} ln ∣ x + 1 ∣

\int \frac{7}{3 ( x - 2 )} d x = \frac{7}{3} ln ∣ x - 2 ∣

= \frac{10}{3} ln ∣ x + 1 ∣ - \frac{7}{3} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{10}{3} ln ∣ x + 1 ∣ - \frac{7}{3} ln ∣ x - 2 ∣ + C

\int 8 d x

\int \frac{18 x ^{2}}{3 x ^{3} + 4} d x

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

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

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