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Beliebt Rechnen >

integral ((x^2))/(((2x+x^3)^3))

Lösung

integral

\frac{( x ^{2} )}{( ( 2 x + x ^{3} ) ^{3} )}

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \int \frac{1}{8 x} - \frac{x}{8 ( x ^{2} + 2 )} - \frac{x}{4 ( x ^{2} + 2 ) ^{2}} - \frac{x}{2 ( x ^{2} + 2 ) ^{3}} 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}{8 x} d x - \int \frac{x}{8 ( x ^{2} + 2 )} d x - \int \frac{x}{4 ( x ^{2} + 2 ) ^{2}} d x - \int \frac{x}{2 ( x ^{2} + 2 ) ^{3}} d x

\int \frac{1}{8 x} d x = \frac{1}{8} ln ∣ x ∣

\int \frac{x}{8 ( x ^{2} + 2 )} d x = \frac{1}{16} ln x^{2} + 2

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

\int \frac{x}{2 ( x ^{2} + 2 ) ^{3}} d x = - \frac{1}{8 ( x ^{2} + 2 ) ^{2}}

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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