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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} ln x^{2} + 2 + \frac{1}{x ^{2} + 2} - \frac{1}{2 ( x ^{2} + 2 )} - \frac{1}{4 2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

\int \frac{1}{( x ^{2} + 2 ) ^{2}} d x = \frac{1}{4 2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2})))

= \frac{1}{2} ln x^{2} + 2 + \frac{1}{x ^{2} + 2} - \frac{1}{2 ( x ^{2} + 2 )} - \frac{1}{4 2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2})))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} + 2 + \frac{1}{x ^{2} + 2} - \frac{1}{2 ( x ^{2} + 2 )} - \frac{1}{4 2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) + C

Graph

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