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Beliebt Rechnen >

integral of (40)/((x^2-x)(x^2+4))

Lösung

\int \frac{40}{( x ^{2} - x ) ( x ^{2} + 4 )} d x

Lösung

40 (- \frac{1}{4} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{20} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 - 2 arctan (\frac{x}{2}))) + C

Schritte zur Lösung

\int \frac{40}{( x ^{2} - x ) ( x ^{2} + 4 )} d x

Entferne die Konstante:

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

= 40 \cdot \int \frac{1}{( x ^{2} - x ) ( x ^{2} + 4 )} d x

Ermittle den Partialbruch von

\frac{1}{( x ^{2} - x ) ( x ^{2} + 4 )} : - \frac{1}{4 x} + \frac{1}{5 ( x - 1 )} + \frac{x - 4}{20 ( x ^{2} + 4 )}

= 40 \cdot \int - \frac{1}{4 x} + \frac{1}{5 ( x - 1 )} + \frac{x - 4}{20 ( x ^{2} + 4 )} 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

= 40 (- \int \frac{1}{4 x} d x + \int \frac{1}{5 ( x - 1 )} d x + \int \frac{x - 4}{20 ( x ^{2} + 4 )} d x)

\int \frac{1}{4 x} d x = \frac{1}{4} ln ∣ x ∣

\int \frac{1}{5 ( x - 1 )} d x = \frac{1}{5} ln ∣ x - 1 ∣

\int \frac{x - 4}{20 ( x ^{2} + 4 )} d x = \frac{1}{20} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 - 2 arctan (\frac{x}{2}))

= 40 (- \frac{1}{4} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{20} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 - 2 arctan (\frac{x}{2})))

Füge eine Konstante zur Lösung hinzu

= 40 (- \frac{1}{4} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{20} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 - 2 arctan (\frac{x}{2}))) + C

Graph

Beliebte Beispiele

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\int \frac{7 e ^{x}}{( e ^{x} - 2 ) ( e ^{2 x} + 3 )} d x

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

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\int \frac{1}{( x + 1 ) ^{2} ( x ^{2} + 2 )} d x

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\int 1 - cos (\frac{1}{x}) d x