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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

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