integral of (9x^2+5)/((4x-1)^7)

Lösung

\int \frac{9 x ^{2} + 5}{( 4 x - 1 ) ^{7}} d x

- \frac{9}{256 ( 4 x - 1 ) ^{4}} - \frac{9}{160 ( 4 x - 1 ) ^{5}} - \frac{89}{384 ( 4 x - 1 ) ^{6}} + C

Schritte zur Lösung

\int \frac{9 x ^{2} + 5}{( 4 x - 1 ) ^{7}} d x

Ermittle den Partialbruch von

\frac{9 x ^{2} + 5}{( 4 x - 1 ) ^{7}} : \frac{9}{16 ( 4 x - 1 ) ^{5}} + \frac{9}{8 ( 4 x - 1 ) ^{6}} + \frac{89}{16 ( 4 x - 1 ) ^{7}}

= \int \frac{9}{16 ( 4 x - 1 ) ^{5}} + \frac{9}{8 ( 4 x - 1 ) ^{6}} + \frac{89}{16 ( 4 x - 1 ) ^{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{9}{16 ( 4 x - 1 ) ^{5}} d x + \int \frac{9}{8 ( 4 x - 1 ) ^{6}} d x + \int \frac{89}{16 ( 4 x - 1 ) ^{7}} d x

\int \frac{9}{16 ( 4 x - 1 ) ^{5}} d x = - \frac{9}{256 ( 4 x - 1 ) ^{4}}

\int \frac{9}{8 ( 4 x - 1 ) ^{6}} d x = - \frac{9}{160 ( 4 x - 1 ) ^{5}}

\int \frac{89}{16 ( 4 x - 1 ) ^{7}} d x = - \frac{89}{384 ( 4 x - 1 ) ^{6}}

= - \frac{9}{256 ( 4 x - 1 ) ^{4}} - \frac{9}{160 ( 4 x - 1 ) ^{5}} - \frac{89}{384 ( 4 x - 1 ) ^{6}}

Füge eine Konstante zur Lösung hinzu

= - \frac{9}{256 ( 4 x - 1 ) ^{4}} - \frac{9}{160 ( 4 x - 1 ) ^{5}} - \frac{89}{384 ( 4 x - 1 ) ^{6}} + C

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