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

Lösung

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

\frac{2}{11} ln ∣ x + 2 ∣ + \frac{1}{55} ln ∣ 5 x - 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x}{( x + 2 ) ( 5 x - 1 )} : \frac{2}{11 ( x + 2 )} + \frac{1}{11 ( 5 x - 1 )}

= \int \frac{2}{11 ( x + 2 )} + \frac{1}{11 ( 5 x - 1 )} 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{2}{11 ( x + 2 )} d x + \int \frac{1}{11 ( 5 x - 1 )} d x

\int \frac{2}{11 ( x + 2 )} d x = \frac{2}{11} ln ∣ x + 2 ∣

\int \frac{1}{11 ( 5 x - 1 )} d x = \frac{1}{55} ln ∣ 5 x - 1 ∣

= \frac{2}{11} ln ∣ x + 2 ∣ + \frac{1}{55} ln ∣ 5 x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{2}{11} ln ∣ x + 2 ∣ + \frac{1}{55} ln ∣ 5 x - 1 ∣ + C