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Beliebt Rechnen >

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

Lösung

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

Lösung

- 2 (- \frac{1}{32 2 ( x + 2 )} - \frac{1}{32 ( x + 2 ) ^{2}} + \frac{1}{32 2 ( x - 2 )} - \frac{1}{32 ( x - 2 ) ^{2}}) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

= - 2 \cdot \int \frac{x}{( x ^{2} - 2 ) ^{3}} d x

Ermittle den Partialbruch von

\frac{x}{( x ^{2} - 2 ) ^{3}} : \frac{1}{32 2 ( x + 2 ) ^{2}} + \frac{1}{16 ( x + 2 ) ^{3}} - \frac{1}{32 2 ( x - 2 ) ^{2}} + \frac{1}{16 ( x - 2 ) ^{3}}

= - 2 \cdot \int \frac{1}{32 2 ( x + 2 ) ^{2}} + \frac{1}{16 ( x + 2 ) ^{3}} - \frac{1}{32 2 ( x - 2 ) ^{2}} + \frac{1}{16 ( x - 2 ) ^{3}} 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}{32 2 ( x + 2 ) ^{2}} d x + \int \frac{1}{16 ( x + 2 ) ^{3}} d x - \int \frac{1}{32 2 ( x - 2 ) ^{2}} d x + \int \frac{1}{16 ( x - 2 ) ^{3}} d x)

\int \frac{1}{32 2 ( x + 2 ) ^{2}} d x = - \frac{1}{32 2 ( x + 2 )}

\int \frac{1}{16 ( x + 2 ) ^{3}} d x = - \frac{1}{32 ( x + 2 ) ^{2}}

\int \frac{1}{32 2 ( x - 2 ) ^{2}} d x = - \frac{1}{32 2 ( x - 2 )}

\int \frac{1}{16 ( x - 2 ) ^{3}} d x = - \frac{1}{32 ( x - 2 ) ^{2}}

= - 2 (- \frac{1}{32 2 ( x + 2 )} - \frac{1}{32 ( x + 2 ) ^{2}} - (- \frac{1}{32 2 ( x - 2 )}) - \frac{1}{32 ( x - 2 ) ^{2}})

Vereinfache

= - 2 (- \frac{1}{32 2 ( x + 2 )} - \frac{1}{32 ( x + 2 ) ^{2}} + \frac{1}{32 2 ( x - 2 )} - \frac{1}{32 ( x - 2 ) ^{2}})

Füge eine Konstante zur Lösung hinzu

= - 2 (- \frac{1}{32 2 ( x + 2 )} - \frac{1}{32 ( x + 2 ) ^{2}} + \frac{1}{32 2 ( x - 2 )} - \frac{1}{32 ( x - 2 ) ^{2}}) + C

Graph

Beliebte Beispiele

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