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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{x ( 5 - x )} : \frac{1}{5 x} - \frac{1}{5 ( x - 5 )}

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

\int \frac{1}{5 x} d x = \frac{1}{5} ln ∣ x ∣

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

= \frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x - 5 ∣

Füge eine Konstante zur Lösung hinzu

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

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