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Beliebt Rechnen >

integral of (3x^2+x)/((8x^3+4x^2)^2)

Lösung

\int \frac{3 x ^{2} + x}{( 8 x ^{3} + 4 x ^{2} ) ^{2}} d x

Lösung

\frac{1}{16 x} - \frac{1}{32 x ^{2}} - \frac{1}{8 ( 2 x + 1 )} + C

Schritte zur Lösung

\int \frac{3 x ^{2} + x}{( 8 x ^{3} + 4 x ^{2} ) ^{2}} d x

Ermittle den Partialbruch von

\frac{3 x ^{2} + x}{( 8 x ^{3} + 4 x ^{2} ) ^{2}} : - \frac{1}{16 x ^{2}} + \frac{1}{16 x ^{3}} + \frac{1}{4 ( 2 x + 1 ) ^{2}}

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

\int \frac{1}{16 x ^{2}} d x = - \frac{1}{16 x}

\int \frac{1}{16 x ^{3}} d x = - \frac{1}{32 x ^{2}}

\int \frac{1}{4 ( 2 x + 1 ) ^{2}} d x = - \frac{1}{8 ( 2 x + 1 )}

= - (- \frac{1}{16 x}) - \frac{1}{32 x ^{2}} - \frac{1}{8 ( 2 x + 1 )}

Vereinfache

= \frac{1}{16 x} - \frac{1}{32 x ^{2}} - \frac{1}{8 ( 2 x + 1 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{16 x} - \frac{1}{32 x ^{2}} - \frac{1}{8 ( 2 x + 1 )} + C

Graph

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