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Beliebt Rechnen >

integral of 2/((x-2)(x^4+4))

Lösung

\int \frac{2}{( x - 2 ) ( x ^{4} + 4 )} d x

Lösung

2 (\frac{1}{20} ln ∣ x - 2 ∣ + \frac{1}{40} (- ln x^{2} + 2 x + 2 - arctan (x + 1)) - \frac{1}{8} arctan (x - 1)) + C

Schritte zur Lösung

\int \frac{2}{( x - 2 ) ( x ^{4} + 4 )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{( x - 2 ) ( x ^{4} + 4 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 2 ) ( x ^{4} + 4 )} : \frac{1}{20 ( x - 2 )} + \frac{- 2 x - 3}{40 ( x ^{2} + 2 x + 2 )} - \frac{1}{8 ( x ^{2} - 2 x + 2 )}

= 2 \cdot \int \frac{1}{20 ( x - 2 )} + \frac{- 2 x - 3}{40 ( x ^{2} + 2 x + 2 )} - \frac{1}{8 ( x ^{2} - 2 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

= 2 (\int \frac{1}{20 ( x - 2 )} d x + \int \frac{- 2 x - 3}{40 ( x ^{2} + 2 x + 2 )} d x - \int \frac{1}{8 ( x ^{2} - 2 x + 2 )} d x)

\int \frac{1}{20 ( x - 2 )} d x = \frac{1}{20} ln ∣ x - 2 ∣

\int \frac{- 2 x - 3}{40 ( x ^{2} + 2 x + 2 )} d x = \frac{1}{40} (- ln x^{2} + 2 x + 2 - arctan (x + 1))

\int \frac{1}{8 ( x ^{2} - 2 x + 2 )} d x = \frac{1}{8} arctan (x - 1)

= 2 (\frac{1}{20} ln ∣ x - 2 ∣ + \frac{1}{40} (- ln x^{2} + 2 x + 2 - arctan (x + 1)) - \frac{1}{8} arctan (x - 1))

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{20} ln ∣ x - 2 ∣ + \frac{1}{40} (- ln x^{2} + 2 x + 2 - arctan (x + 1)) - \frac{1}{8} arctan (x - 1)) + C

Graph

Beliebte Beispiele

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