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Beliebt Rechnen >

integral of ((x^3-4x))/((x^2+1)^2)

Lösung

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

Lösung

\frac{1}{2} ln x^{2} + 1 + \frac{5}{2 ( x ^{2} + 1 )} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{3} - 4 x}{( x ^{2} + 1 ) ^{2}} : \frac{x}{x ^{2} + 1} - \frac{5 x}{( x ^{2} + 1 ) ^{2}}

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

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

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

= \frac{1}{2} ln x^{2} + 1 - (- \frac{5}{2 ( x ^{2} + 1 )})

Vereinfache

= \frac{1}{2} ln x^{2} + 1 + \frac{5}{2 ( x ^{2} + 1 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} + 1 + \frac{5}{2 ( x ^{2} + 1 )} + C

Graph

Beliebte Beispiele

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x \to \infty + lim (\frac{1}{x})

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\int \frac{1}{6} sec (x) tan (x) d x

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\frac{d}{d x} ((a - \frac{b}{x})^{2})

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