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Beliebt Rechnen >

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

Lösung

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

Lösung

- ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

= ln ∣ x - 1 ∣ - 2 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

Vereinfache

ln ∣ x - 1 ∣ - 2 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) : - ln ∣ x - 1 ∣ + 2 ln ∣ x ∣

= - ln ∣ x - 1 ∣ + 2 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral of x(4-x^2)

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

integral of 6/(cos(θ)-1)

\int \frac{6}{cos ( θ ) - 1} d θ

integral of x^3(9+x^4)^7

\int x^{3} (9 + x^{4})^{7} d x

integral of (sinh(x))/(cosh^3(x))

\int \frac{sinh ( x )}{cosh ^{3} ( x )} d x

integral of sqrt(x)-2/(x^5)+7/x

\int x - \frac{2}{x ^{5}} + \frac{7}{x} d x