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Beliebt Rechnen >

integral of (x^2-x+12)/(x^3+2x)

Lösung

\int \frac{x ^{2} - x + 12}{x ^{3} + 2 x} d x

Lösung

6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2}) + C

Schritte zur Lösung

\int \frac{x ^{2} - x + 12}{x ^{3} + 2 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} - x + 12}{x ^{3} + 2 x} : \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 2}

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

\int \frac{6}{x} d x = 6 ln ∣ x ∣

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

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2})

Füge eine Konstante zur Lösung hinzu

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2}) + C

Graph

Beliebte Beispiele

integral of (x+5)/x

\int \frac{x + 5}{x} d x

derivative of 1/3 sin(x)

\frac{d}{d x} (\frac{1}{3} sin (x))

derivative f(x)=(3/5 x)^9

d er i v a t i v e f (x) = (\frac{3}{5} x)^{9}

derivative of (21/(64x^{11/4)})

\frac{d}{d x} (\frac{21}{64 x ^{\frac{11}{4}}})

integral from 0 to 1 of x^2((2(x+1))/3)

\int_{0}^{1} x^{2} (\frac{2 ( x + 1 )}{3}) d x