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Beliebt Rechnen >

integral of (2x^2-1)/(x^3-5x^2+6x)

Lösung

\int \frac{2 x ^{2} - 1}{x ^{3} - 5 x ^{2} + 6 x} d x

Lösung

- \frac{1}{6} ln ∣ x ∣ - \frac{7}{2} ln ∣ x - 2 ∣ + \frac{17}{3} ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{2 x ^{2} - 1}{x ^{3} - 5 x ^{2} + 6 x} d x

Multipliziere aus

\frac{2 x ^{2} - 1}{x ^{3} - 5 x ^{2} + 6 x} : \frac{2 x ^{2}}{x ^{3} - 5 x ^{2} + 6 x} - \frac{1}{x ^{3} - 5 x ^{2} + 6 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{2 x ^{2}}{x ^{3} - 5 x ^{2} + 6 x} d x - \int \frac{1}{x ^{3} - 5 x ^{2} + 6 x} d x

\int \frac{2 x ^{2}}{x ^{3} - 5 x ^{2} + 6 x} d x = 2 (- 2 ln ∣ x - 2 ∣ + 3 ln ∣ x - 3 ∣)

\int \frac{1}{x ^{3} - 5 x ^{2} + 6 x} d x = \frac{1}{6} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x - 3 ∣

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

Vereinfache

2 (- 2 ln ∣ x - 2 ∣ + 3 ln ∣ x - 3 ∣) - (\frac{1}{6} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x - 3 ∣) : - \frac{1}{6} ln ∣ x ∣ - \frac{7}{2} ln ∣ x - 2 ∣ + \frac{17}{3} ln ∣ x - 3 ∣

= - \frac{1}{6} ln ∣ x ∣ - \frac{7}{2} ln ∣ x - 2 ∣ + \frac{17}{3} ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{6} ln ∣ x ∣ - \frac{7}{2} ln ∣ x - 2 ∣ + \frac{17}{3} ln ∣ x - 3 ∣ + C

Graph

Beliebte Beispiele

integral of (x^2-6x)^5(x-3)

\int (x^{2} - 6 x)^{5} (x - 3) d x

integral of x^{-3}+2x-6x^2+cos(x)

\int x^{- 3} + 2 x - 6 x^{2} + cos (x) d x

integral of xsqrt((x+3))

\int x (x + 3) d x

integral of (3-2x)/(sqrt(1-x^2))

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

integral of (4x^5+5x^3+11)

\int (4 x^{5} + 5 x^{3} + 11) d x