integral of (5x-4)/(x^2-4x)

Lösung

\int \frac{5 x - 4}{x ^{2} - 4 x} d x

ln ∣ x ∣ + 4 ln ∣ x - 4 ∣ + C

Schritte zur Lösung

\int \frac{5 x - 4}{x ^{2} - 4 x} d x

Ermittle den Partialbruch von

\frac{5 x - 4}{x ^{2} - 4 x} : \frac{1}{x} + \frac{4}{x - 4}

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{4}{x - 4} d x = 4 ln ∣ x - 4 ∣

= ln ∣ x ∣ + 4 ln ∣ x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + 4 ln ∣ x - 4 ∣ + C

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