integral of 1/((x+4)(x+8))

Lösung

\int \frac{1}{( x + 4 ) ( x + 8 )} d x

\frac{1}{4} ln ∣ x + 4 ∣ - \frac{1}{4} ln ∣ x + 8 ∣ + C

Schritte zur Lösung

\int \frac{1}{( x + 4 ) ( x + 8 )} d x

Ermittle den Partialbruch von

\frac{1}{( x + 4 ) ( x + 8 )} : \frac{1}{4 ( x + 4 )} - \frac{1}{4 ( x + 8 )}

= \int \frac{1}{4 ( x + 4 )} - \frac{1}{4 ( x + 8 )} 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}{4 ( x + 4 )} d x - \int \frac{1}{4 ( x + 8 )} d x

\int \frac{1}{4 ( x + 4 )} d x = \frac{1}{4} ln ∣ x + 4 ∣

\int \frac{1}{4 ( x + 8 )} d x = \frac{1}{4} ln ∣ x + 8 ∣

= \frac{1}{4} ln ∣ x + 4 ∣ - \frac{1}{4} ln ∣ x + 8 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} ln ∣ x + 4 ∣ - \frac{1}{4} ln ∣ x + 8 ∣ + C