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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x - 1}{x ^{3} + x ^{2} - 6 x} : \frac{1}{6 x} + \frac{1}{10 ( x - 2 )} - \frac{4}{15 ( x + 3 )}

= \int \frac{1}{6 x} + \frac{1}{10 ( x - 2 )} - \frac{4}{15 ( x + 3 )} d 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{1}{6 x} d x + \int \frac{1}{10 ( x - 2 )} d x - \int \frac{4}{15 ( x + 3 )} d x

\int \frac{1}{6 x} d x = \frac{1}{6} ln ∣ x ∣

\int \frac{1}{10 ( x - 2 )} d x = \frac{1}{10} ln ∣ x - 2 ∣

\int \frac{4}{15 ( x + 3 )} d x = \frac{4}{15} ln ∣ x + 3 ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 60x^{-1/5}tan(x^{4/5})

\int 60 x^{- \frac{1}{5}} tan (x^{\frac{4}{5}}) d x

integral of sqrt(-ln(x))

\int - ln (x) d x

integral of 1/((x-1)^5)

\int \frac{1}{( x - 1 ) ^{5}} d x

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

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

integral of 1/(x^3y^2)

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