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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{10} ln ∣ x + 2 ∣ + \frac{4}{15} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{1}{x ^{3} - x ^{2} - 6 x} d x = - \frac{1}{6} ln ∣ x ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣

= - \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣

Vereinfache

- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣ : - \frac{1}{10} ln ∣ x + 2 ∣ + \frac{4}{15} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x ∣

= - \frac{1}{10} ln ∣ x + 2 ∣ + \frac{4}{15} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{10} ln ∣ x + 2 ∣ + \frac{4}{15} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral of x^2+\sqrt[4]{x}

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

integral of 3/(sqrt(e^{2a)-4)}

\int \frac{3}{e ^{2 a} - 4} d a

integral of 4/(x^2(2x+3))

\int \frac{4}{x ^{2} ( 2 x + 3 )} d x

integral of 1/(x^2-4x+20)

\int \frac{1}{x ^{2} - 4 x + 20} d x

integral of (28x^3-56x^2+9)/(x^2-2x)

\int \frac{28 x ^{3} - 56 x ^{2} + 9}{x ^{2} - 2 x} d x