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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{40}{x} - \frac{10}{x ^{2}} + \frac{2}{x ^{3}} - \frac{1}{4 x ^{4}} - \frac{80}{2 x + 1} - \frac{40}{( 2 x + 1 ) ^{2}} - \frac{16}{( 2 x + 1 ) ^{3}} - \frac{4}{( 2 x + 1 ) ^{4}} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{4 x + 1}{( 2 x ^{2} + x ) ^{5}} : - \frac{40}{x ^{2}} + \frac{20}{x ^{3}} - \frac{6}{x ^{4}} + \frac{1}{x ^{5}} + \frac{160}{( 2 x + 1 ) ^{2}} + \frac{160}{( 2 x + 1 ) ^{3}} + \frac{96}{( 2 x + 1 ) ^{4}} + \frac{32}{( 2 x + 1 ) ^{5}}

= \int - \frac{40}{x ^{2}} + \frac{20}{x ^{3}} - \frac{6}{x ^{4}} + \frac{1}{x ^{5}} + \frac{160}{( 2 x + 1 ) ^{2}} + \frac{160}{( 2 x + 1 ) ^{3}} + \frac{96}{( 2 x + 1 ) ^{4}} + \frac{32}{( 2 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{40}{x ^{2}} d x + \int \frac{20}{x ^{3}} d x - \int \frac{6}{x ^{4}} d x + \int \frac{1}{x ^{5}} d x + \int \frac{160}{( 2 x + 1 ) ^{2}} d x + \int \frac{160}{( 2 x + 1 ) ^{3}} d x + \int \frac{96}{( 2 x + 1 ) ^{4}} d x + \int \frac{32}{( 2 x + 1 ) ^{5}} d x

\int \frac{40}{x ^{2}} d x = - \frac{40}{x}

\int \frac{20}{x ^{3}} d x = - \frac{10}{x ^{2}}

\int \frac{6}{x ^{4}} d x = - \frac{2}{x ^{3}}

\int \frac{1}{x ^{5}} d x = - \frac{1}{4 x ^{4}}

\int \frac{160}{( 2 x + 1 ) ^{2}} d x = - \frac{80}{2 x + 1}

\int \frac{160}{( 2 x + 1 ) ^{3}} d x = - \frac{40}{( 2 x + 1 ) ^{2}}

\int \frac{96}{( 2 x + 1 ) ^{4}} d x = - \frac{16}{( 2 x + 1 ) ^{3}}

\int \frac{32}{( 2 x + 1 ) ^{5}} d x = - \frac{4}{( 2 x + 1 ) ^{4}}

= - (- \frac{40}{x}) - \frac{10}{x ^{2}} - (- \frac{2}{x ^{3}}) - \frac{1}{4 x ^{4}} - \frac{80}{2 x + 1} - \frac{40}{( 2 x + 1 ) ^{2}} - \frac{16}{( 2 x + 1 ) ^{3}} - \frac{4}{( 2 x + 1 ) ^{4}}

Vereinfache

= \frac{40}{x} - \frac{10}{x ^{2}} + \frac{2}{x ^{3}} - \frac{1}{4 x ^{4}} - \frac{80}{2 x + 1} - \frac{40}{( 2 x + 1 ) ^{2}} - \frac{16}{( 2 x + 1 ) ^{3}} - \frac{4}{( 2 x + 1 ) ^{4}}

Füge eine Konstante zur Lösung hinzu

= \frac{40}{x} - \frac{10}{x ^{2}} + \frac{2}{x ^{3}} - \frac{1}{4 x ^{4}} - \frac{80}{2 x + 1} - \frac{40}{( 2 x + 1 ) ^{2}} - \frac{16}{( 2 x + 1 ) ^{3}} - \frac{4}{( 2 x + 1 ) ^{4}} + C

Graph

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