integral of 1/(x^2+7x+10)

Lösung

\int \frac{1}{x ^{2} + 7 x + 10} d x

\frac{1}{3} ln ∣ x + 2 ∣ - \frac{1}{3} ln ∣ x + 5 ∣ + C

Schritte zur Lösung

\int \frac{1}{x ^{2} + 7 x + 10} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} + 7 x + 10} : \frac{1}{3 ( x + 2 )} - \frac{1}{3 ( x + 5 )}

= \int \frac{1}{3 ( x + 2 )} - \frac{1}{3 ( x + 5 )} 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}{3 ( x + 2 )} d x - \int \frac{1}{3 ( x + 5 )} d x

\int \frac{1}{3 ( x + 2 )} d x = \frac{1}{3} ln ∣ x + 2 ∣

\int \frac{1}{3 ( x + 5 )} d x = \frac{1}{3} ln ∣ x + 5 ∣

= \frac{1}{3} ln ∣ x + 2 ∣ - \frac{1}{3} ln ∣ x + 5 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln ∣ x + 2 ∣ - \frac{1}{3} ln ∣ x + 5 ∣ + C

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