integral of (x^2-4x-1)/(x(x-1))

Lösung

\int \frac{x ^{2} - 4 x - 1}{x ( x - 1 )} d x

x - 4 ln ∣ x - 1 ∣ + ln ∣ x ∣ - 1 + C

Schritte zur Lösung

\int \frac{x ^{2} - 4 x - 1}{x ( x - 1 )} d x

Multipliziere aus

\frac{x ^{2} - 4 x - 1}{x ( x - 1 )} : \frac{x ^{2}}{x ( x - 1 )} - \frac{4 x}{x ( x - 1 )} - \frac{1}{x ( x - 1 )}

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{x ^{2}}{x ( x - 1 )} d x - \int \frac{4 x}{x ( x - 1 )} d x - \int \frac{1}{x ( x - 1 )} d x

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

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

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

= x - 1 + ln ∣ x - 1 ∣ - 4 ln ∣ x - 1 ∣ - (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

Vereinfache

= x - 4 ln ∣ x - 1 ∣ + ln ∣ x ∣ - 1

Füge eine Konstante zur Lösung hinzu

= x - 4 ln ∣ x - 1 ∣ + ln ∣ x ∣ - 1 + C

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