integral of x/((x-1)(x^2+6x+11))

Lösung

\int \frac{x}{( x - 1 ) ( x ^{2} + 6 x + 11 )} d x

\frac{1}{18} ln ∣ x - 1 ∣ + \frac{1}{36} (- ln x^{2} + 6 x + 11 + 142 arctan (\frac{1}{2} (x + 3))) + C

Schritte zur Lösung

\int \frac{x}{( x - 1 ) ( x ^{2} + 6 x + 11 )} d x

Ermittle den Partialbruch von

\frac{x}{( x - 1 ) ( x ^{2} + 6 x + 11 )} : \frac{1}{18 ( x - 1 )} + \frac{- x + 11}{18 ( x ^{2} + 6 x + 11 )}

= \int \frac{1}{18 ( x - 1 )} + \frac{- x + 11}{18 ( x ^{2} + 6 x + 11 )} 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}{18 ( x - 1 )} d x + \int \frac{- x + 11}{18 ( x ^{2} + 6 x + 11 )} d x

\int \frac{1}{18 ( x - 1 )} d x = \frac{1}{18} ln ∣ x - 1 ∣

\int \frac{- x + 11}{18 ( x ^{2} + 6 x + 11 )} d x = \frac{1}{36} (- ln x^{2} + 6 x + 11 + 142 arctan (\frac{1}{2} (x + 3)))

= \frac{1}{18} ln ∣ x - 1 ∣ + \frac{1}{36} (- ln x^{2} + 6 x + 11 + 142 arctan (\frac{1}{2} (x + 3)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{18} ln ∣ x - 1 ∣ + \frac{1}{36} (- ln x^{2} + 6 x + 11 + 142 arctan (\frac{1}{2} (x + 3))) + C

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