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Beliebt Rechnen >

integral of 1/(x^3-x^2-x-15)

Lösung

\int \frac{1}{x ^{3} - x ^{2} - x - 15} d x

Lösung

\frac{1}{20} ln ∣ x - 3 ∣ + \frac{1}{40} (- ln x^{2} + 2 x + 5 - 4 arctan (\frac{1}{2} (x + 1))) + C

Schritte zur Lösung

\int \frac{1}{x ^{3} - x ^{2} - x - 15} d x

Ermittle den Partialbruch von

\frac{1}{x ^{3} - x ^{2} - x - 15} : \frac{1}{20 ( x - 3 )} + \frac{- x - 5}{20 ( x ^{2} + 2 x + 5 )}

= \int \frac{1}{20 ( x - 3 )} + \frac{- x - 5}{20 ( x ^{2} + 2 x + 5 )} 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}{20 ( x - 3 )} d x + \int \frac{- x - 5}{20 ( x ^{2} + 2 x + 5 )} d x

\int \frac{1}{20 ( x - 3 )} d x = \frac{1}{20} ln ∣ x - 3 ∣

\int \frac{- x - 5}{20 ( x ^{2} + 2 x + 5 )} d x = \frac{1}{40} (- ln x^{2} + 2 x + 5 - 4 arctan (\frac{1}{2} (x + 1)))

= \frac{1}{20} ln ∣ x - 3 ∣ + \frac{1}{40} (- ln x^{2} + 2 x + 5 - 4 arctan (\frac{1}{2} (x + 1)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{20} ln ∣ x - 3 ∣ + \frac{1}{40} (- ln x^{2} + 2 x + 5 - 4 arctan (\frac{1}{2} (x + 1))) + C

Graph

Beliebte Beispiele

integral of x^{-3}ln(-5x)

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

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

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

integral of 1/2 tan(x)

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

integral of (sin(5x))/(cos^4(5x))

\int \frac{sin ( 5 x )}{cos ^{4} ( 5 x )} d x

integral of ln(3)x^4*2x^5

\int ln (3) x^{4} \cdot 2 x^{5} d x