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Beliebt Rechnen >

integral of (-2x^2+x-2)/(x^5+2x^3+x)

Lösung

\int \frac{- 2 x ^{2} + x - 2}{x ^{5} + 2 x ^{3} + x} d x

Lösung

\frac{x}{x ^{4} + 2 x ^{2} + 1} + \frac{1}{2} arctan (x) - \frac{1}{8} sin (4 arctan (x)) - 2 ln ∣ x ∣ + ln x^{2} + 1 + C

Schritte zur Lösung

\int \frac{- 2 x ^{2} + x - 2}{x ^{5} + 2 x ^{3} + x} d x

Multipliziere aus

\frac{- 2 x ^{2} + x - 2}{x ^{5} + 2 x ^{3} + x} : - \frac{2 x ^{2}}{x ^{5} + 2 x ^{3} + x} + \frac{x}{x ^{5} + 2 x ^{3} + x} - \frac{2}{x ^{5} + 2 x ^{3} + 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 ^{2}}{x ^{5} + 2 x ^{3} + x} d x + \int \frac{x}{x ^{5} + 2 x ^{3} + x} d x - \int \frac{2}{x ^{5} + 2 x ^{3} + x} d x

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

\int \frac{x}{x ^{5} + 2 x ^{3} + x} d x = \frac{x}{x ^{4} + 2 x ^{2} + 1} + \frac{1}{2} (arctan (x) - \frac{1}{4} sin (4 arctan (x)))

\int \frac{2}{x ^{5} + 2 x ^{3} + x} d x = 2 ln ∣ x ∣ - ln x^{2} + 1 + \frac{1}{x ^{2} + 1}

= - (- \frac{1}{x ^{2} + 1}) + \frac{x}{x ^{4} + 2 x ^{2} + 1} + \frac{1}{2} (arctan (x) - \frac{1}{4} sin (4 arctan (x))) - (2 ln ∣ x ∣ - ln x^{2} + 1 + \frac{1}{x ^{2} + 1})

Vereinfache

- (- \frac{1}{x ^{2} + 1}) + \frac{x}{x ^{4} + 2 x ^{2} + 1} + \frac{1}{2} (arctan (x) - \frac{1}{4} sin (4 arctan (x))) - (2 ln ∣ x ∣ - ln x^{2} + 1 + \frac{1}{x ^{2} + 1}) : \frac{x}{x ^{4} + 2 x ^{2} + 1} + \frac{1}{2} arctan (x) - \frac{1}{8} sin (4 arctan (x)) - 2 ln ∣ x ∣ + ln x^{2} + 1

= \frac{x}{x ^{4} + 2 x ^{2} + 1} + \frac{1}{2} arctan (x) - \frac{1}{8} sin (4 arctan (x)) - 2 ln ∣ x ∣ + ln x^{2} + 1

Füge eine Konstante zur Lösung hinzu

= \frac{x}{x ^{4} + 2 x ^{2} + 1} + \frac{1}{2} arctan (x) - \frac{1}{8} sin (4 arctan (x)) - 2 ln ∣ x ∣ + ln x^{2} + 1 + C

Graph

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