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Beliebt Rechnen >

integral of (x^7)/((1-x^2)^5)

Lösung

\int \frac{x ^{7}}{( 1 - x ^{2} ) ^{5}} d x

Lösung

- \frac{35}{256 ( x + 1 )} + \frac{29}{256 ( x + 1 ) ^{2}} - \frac{3}{64 ( x + 1 ) ^{3}} + \frac{1}{128 ( x + 1 ) ^{4}} + \frac{35}{256 ( x - 1 )} + \frac{29}{256 ( x - 1 ) ^{2}} + \frac{3}{64 ( x - 1 ) ^{3}} + \frac{1}{128 ( x - 1 ) ^{4}} + C

Schritte zur Lösung

\int \frac{x ^{7}}{( 1 - x ^{2} ) ^{5}} d x

Ermittle den Partialbruch von

\frac{x ^{7}}{( 1 - x ^{2} ) ^{5}} : \frac{35}{256 ( x + 1 ) ^{2}} - \frac{29}{128 ( x + 1 ) ^{3}} + \frac{9}{64 ( x + 1 ) ^{4}} - \frac{1}{32 ( x + 1 ) ^{5}} - \frac{35}{256 ( x - 1 ) ^{2}} - \frac{29}{128 ( x - 1 ) ^{3}} - \frac{9}{64 ( x - 1 ) ^{4}} - \frac{1}{32 ( x - 1 ) ^{5}}

= \int \frac{35}{256 ( x + 1 ) ^{2}} - \frac{29}{128 ( x + 1 ) ^{3}} + \frac{9}{64 ( x + 1 ) ^{4}} - \frac{1}{32 ( x + 1 ) ^{5}} - \frac{35}{256 ( x - 1 ) ^{2}} - \frac{29}{128 ( x - 1 ) ^{3}} - \frac{9}{64 ( x - 1 ) ^{4}} - \frac{1}{32 ( x - 1 ) ^{5}} 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{35}{256 ( x + 1 ) ^{2}} d x - \int \frac{29}{128 ( x + 1 ) ^{3}} d x + \int \frac{9}{64 ( x + 1 ) ^{4}} d x - \int \frac{1}{32 ( x + 1 ) ^{5}} d x - \int \frac{35}{256 ( x - 1 ) ^{2}} d x - \int \frac{29}{128 ( x - 1 ) ^{3}} d x - \int \frac{9}{64 ( x - 1 ) ^{4}} d x - \int \frac{1}{32 ( x - 1 ) ^{5}} d x

\int \frac{35}{256 ( x + 1 ) ^{2}} d x = - \frac{35}{256 ( x + 1 )}

\int \frac{29}{128 ( x + 1 ) ^{3}} d x = - \frac{29}{256 ( x + 1 ) ^{2}}

\int \frac{9}{64 ( x + 1 ) ^{4}} d x = - \frac{3}{64 ( x + 1 ) ^{3}}

\int \frac{1}{32 ( x + 1 ) ^{5}} d x = - \frac{1}{128 ( x + 1 ) ^{4}}

\int \frac{35}{256 ( x - 1 ) ^{2}} d x = - \frac{35}{256 ( x - 1 )}

\int \frac{29}{128 ( x - 1 ) ^{3}} d x = - \frac{29}{256 ( x - 1 ) ^{2}}

\int \frac{9}{64 ( x - 1 ) ^{4}} d x = - \frac{3}{64 ( x - 1 ) ^{3}}

\int \frac{1}{32 ( x - 1 ) ^{5}} d x = - \frac{1}{128 ( x - 1 ) ^{4}}

= - \frac{35}{256 ( x + 1 )} - (- \frac{29}{256 ( x + 1 ) ^{2}}) - \frac{3}{64 ( x + 1 ) ^{3}} - (- \frac{1}{128 ( x + 1 ) ^{4}}) - (- \frac{35}{256 ( x - 1 )}) - (- \frac{29}{256 ( x - 1 ) ^{2}}) - (- \frac{3}{64 ( x - 1 ) ^{3}}) - (- \frac{1}{128 ( x - 1 ) ^{4}})

Vereinfache

= - \frac{35}{256 ( x + 1 )} + \frac{29}{256 ( x + 1 ) ^{2}} - \frac{3}{64 ( x + 1 ) ^{3}} + \frac{1}{128 ( x + 1 ) ^{4}} + \frac{35}{256 ( x - 1 )} + \frac{29}{256 ( x - 1 ) ^{2}} + \frac{3}{64 ( x - 1 ) ^{3}} + \frac{1}{128 ( x - 1 ) ^{4}}

Füge eine Konstante zur Lösung hinzu

= - \frac{35}{256 ( x + 1 )} + \frac{29}{256 ( x + 1 ) ^{2}} - \frac{3}{64 ( x + 1 ) ^{3}} + \frac{1}{128 ( x + 1 ) ^{4}} + \frac{35}{256 ( x - 1 )} + \frac{29}{256 ( x - 1 ) ^{2}} + \frac{3}{64 ( x - 1 ) ^{3}} + \frac{1}{128 ( x - 1 ) ^{4}} + C

Graph

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