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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{2}{x ^{2} + 2} d x = 2 arctan (\frac{x}{2})

\int \frac{1}{x - 2} d x = ln ∣ x - 2 ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x-1)/(3x^2-6x+5)

\int \frac{x - 1}{3 x ^{2} - 6 x + 5} d x

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\int \frac{1}{x + 45} d x

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\int (csc (4 x) + cot (4 x))^{2} d x

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\int e^{3 x} \cdot cos (5 x) d x

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