integral of 1/(x^2-5x+6)

Lösung

\int \frac{1}{x ^{2} - 5 x + 6} d x

- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{1}{x ^{2} - 5 x + 6} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} - 5 x + 6} : - \frac{1}{x - 2} + \frac{1}{x - 3}

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

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

\int \frac{1}{x - 3} d x = ln ∣ x - 3 ∣

= - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ + C

x \to - 2 lim (\frac{x + 2}{x - 2})

x \to \infty lim (\frac{7}{x})

t an g e n t \frac{3 x + 1}{x ^{2} + 1}

\int \frac{272 x ^{2} + 4}{x ^{2} ( x - 4 )} d x

(2 x y - 3 x^{2}) d x + (x^{2} - 2 y) d y = 0