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Beliebt Rechnen >

integral of (x^3)/((1+2x^2)^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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