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Beliebt Rechnen >

integral of (6x+5)/((3x^2+5x)^6)

Lösung

\int \frac{6 x + 5}{( 3 x ^{2} + 5 x ) ^{6}} d x

Lösung

- \frac{1134}{1953125 x} + \frac{189}{390625 x ^{2}} - \frac{27}{78125 x ^{3}} + \frac{3}{15625 x ^{4}} - \frac{1}{15625 x ^{5}} + \frac{3402}{1953125 ( 3 x + 5 )} + \frac{1701}{390625 ( 3 x + 5 ) ^{2}} + \frac{729}{78125 ( 3 x + 5 ) ^{3}} + \frac{243}{15625 ( 3 x + 5 ) ^{4}} + \frac{243}{15625 ( 3 x + 5 ) ^{5}} + C

Schritte zur Lösung

\int \frac{6 x + 5}{( 3 x ^{2} + 5 x ) ^{6}} d x

Ermittle den Partialbruch von

\frac{6 x + 5}{( 3 x ^{2} + 5 x ) ^{6}} : \frac{1134}{1953125 x ^{2}} - \frac{378}{390625 x ^{3}} + \frac{81}{78125 x ^{4}} - \frac{12}{15625 x ^{5}} + \frac{1}{3125 x ^{6}} - \frac{10206}{1953125 ( 3 x + 5 ) ^{2}} - \frac{10206}{390625 ( 3 x + 5 ) ^{3}} - \frac{6561}{78125 ( 3 x + 5 ) ^{4}} - \frac{2916}{15625 ( 3 x + 5 ) ^{5}} - \frac{729}{3125 ( 3 x + 5 ) ^{6}}

= \int \frac{1134}{1953125 x ^{2}} - \frac{378}{390625 x ^{3}} + \frac{81}{78125 x ^{4}} - \frac{12}{15625 x ^{5}} + \frac{1}{3125 x ^{6}} - \frac{10206}{1953125 ( 3 x + 5 ) ^{2}} - \frac{10206}{390625 ( 3 x + 5 ) ^{3}} - \frac{6561}{78125 ( 3 x + 5 ) ^{4}} - \frac{2916}{15625 ( 3 x + 5 ) ^{5}} - \frac{729}{3125 ( 3 x + 5 ) ^{6}} 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{1134}{1953125 x ^{2}} d x - \int \frac{378}{390625 x ^{3}} d x + \int \frac{81}{78125 x ^{4}} d x - \int \frac{12}{15625 x ^{5}} d x + \int \frac{1}{3125 x ^{6}} d x - \int \frac{10206}{1953125 ( 3 x + 5 ) ^{2}} d x - \int \frac{10206}{390625 ( 3 x + 5 ) ^{3}} d x - \int \frac{6561}{78125 ( 3 x + 5 ) ^{4}} d x - \int \frac{2916}{15625 ( 3 x + 5 ) ^{5}} d x - \int \frac{729}{3125 ( 3 x + 5 ) ^{6}} d x

\int \frac{1134}{1953125 x ^{2}} d x = - \frac{1134}{1953125 x}

\int \frac{378}{390625 x ^{3}} d x = - \frac{189}{390625 x ^{2}}

\int \frac{81}{78125 x ^{4}} d x = - \frac{27}{78125 x ^{3}}

\int \frac{12}{15625 x ^{5}} d x = - \frac{3}{15625 x ^{4}}

\int \frac{1}{3125 x ^{6}} d x = - \frac{1}{15625 x ^{5}}

\int \frac{10206}{1953125 ( 3 x + 5 ) ^{2}} d x = - \frac{3402}{1953125 ( 3 x + 5 )}

\int \frac{10206}{390625 ( 3 x + 5 ) ^{3}} d x = - \frac{1701}{390625 ( 3 x + 5 ) ^{2}}

\int \frac{6561}{78125 ( 3 x + 5 ) ^{4}} d x = - \frac{729}{78125 ( 3 x + 5 ) ^{3}}

\int \frac{2916}{15625 ( 3 x + 5 ) ^{5}} d x = - \frac{243}{15625 ( 3 x + 5 ) ^{4}}

\int \frac{729}{3125 ( 3 x + 5 ) ^{6}} d x = - \frac{243}{15625 ( 3 x + 5 ) ^{5}}

= - \frac{1134}{1953125 x} - (- \frac{189}{390625 x ^{2}}) - \frac{27}{78125 x ^{3}} - (- \frac{3}{15625 x ^{4}}) - \frac{1}{15625 x ^{5}} - (- \frac{3402}{1953125 ( 3 x + 5 )}) - (- \frac{1701}{390625 ( 3 x + 5 ) ^{2}}) - (- \frac{729}{78125 ( 3 x + 5 ) ^{3}}) - (- \frac{243}{15625 ( 3 x + 5 ) ^{4}}) - (- \frac{243}{15625 ( 3 x + 5 ) ^{5}})

Vereinfache

= - \frac{1134}{1953125 x} + \frac{189}{390625 x ^{2}} - \frac{27}{78125 x ^{3}} + \frac{3}{15625 x ^{4}} - \frac{1}{15625 x ^{5}} + \frac{3402}{1953125 ( 3 x + 5 )} + \frac{1701}{390625 ( 3 x + 5 ) ^{2}} + \frac{729}{78125 ( 3 x + 5 ) ^{3}} + \frac{243}{15625 ( 3 x + 5 ) ^{4}} + \frac{243}{15625 ( 3 x + 5 ) ^{5}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1134}{1953125 x} + \frac{189}{390625 x ^{2}} - \frac{27}{78125 x ^{3}} + \frac{3}{15625 x ^{4}} - \frac{1}{15625 x ^{5}} + \frac{3402}{1953125 ( 3 x + 5 )} + \frac{1701}{390625 ( 3 x + 5 ) ^{2}} + \frac{729}{78125 ( 3 x + 5 ) ^{3}} + \frac{243}{15625 ( 3 x + 5 ) ^{4}} + \frac{243}{15625 ( 3 x + 5 ) ^{5}} + C

Graph

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