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Beliebt Rechnen >

integral of (6x^2-3x+1)/(x^3-x^2)

Lösung

\int \frac{6 x ^{2} - 3 x + 1}{x ^{3} - x ^{2}} d x

Lösung

4 ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + \frac{1}{x} + C

Schritte zur Lösung

\int \frac{6 x ^{2} - 3 x + 1}{x ^{3} - x ^{2}} d x

Multipliziere aus

\frac{6 x ^{2} - 3 x + 1}{x ^{3} - x ^{2}} : \frac{6 x ^{2}}{x ^{3} - x ^{2}} - \frac{3 x}{x ^{3} - x ^{2}} + \frac{1}{x ^{3} - x ^{2}}

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{6 x ^{2}}{x ^{3} - x ^{2}} d x - \int \frac{3 x}{x ^{3} - x ^{2}} d x + \int \frac{1}{x ^{3} - x ^{2}} d x

\int \frac{6 x ^{2}}{x ^{3} - x ^{2}} d x = 6 ln ∣ x - 1 ∣

\int \frac{3 x}{x ^{3} - x ^{2}} d x = 3 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

\int \frac{1}{x ^{3} - x ^{2}} d x = - ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣

= 6 ln ∣ x - 1 ∣ - 3 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) - ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣

Vereinfache

6 ln ∣ x - 1 ∣ - 3 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) - ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣ : 4 ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + \frac{1}{x}

= 4 ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + \frac{1}{x}

Füge eine Konstante zur Lösung hinzu

= 4 ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + \frac{1}{x} + C

Graph

Beliebte Beispiele

integral of (1+cot(2θ))^2

\int (1 + cot (2 θ))^{2} d θ

integral of 1/2 (x-2)

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

integral of (x^2)/(sqrt(25-4x^2))

\int \frac{x ^{2}}{25 - 4 x ^{2}} d x

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

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

integral of sqrt(1+4*x^2)

\int 1 + 4 \cdot x^{2} d x