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Beliebt Rechnen >

integral of x/((x+2)^7)

Lösung

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

Lösung

- \frac{1}{5 ( x + 2 ) ^{5}} + \frac{1}{3 ( x + 2 ) ^{6}} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x}{( x + 2 ) ^{7}} : \frac{1}{( x + 2 ) ^{6}} - \frac{2}{( x + 2 ) ^{7}}

= \int \frac{1}{( x + 2 ) ^{6}} - \frac{2}{( x + 2 ) ^{7}} 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{1}{( x + 2 ) ^{6}} d x - \int \frac{2}{( x + 2 ) ^{7}} d x

\int \frac{1}{( x + 2 ) ^{6}} d x = - \frac{1}{5 ( x + 2 ) ^{5}}

\int \frac{2}{( x + 2 ) ^{7}} d x = - \frac{1}{3 ( x + 2 ) ^{6}}

= - \frac{1}{5 ( x + 2 ) ^{5}} - (- \frac{1}{3 ( x + 2 ) ^{6}})

Vereinfache

= - \frac{1}{5 ( x + 2 ) ^{5}} + \frac{1}{3 ( x + 2 ) ^{6}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{5 ( x + 2 ) ^{5}} + \frac{1}{3 ( x + 2 ) ^{6}} + C

Graph

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