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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} - x}{1 + x ^{2}} : \frac{x ^{2}}{1 + x ^{2}} - \frac{x}{1 + x ^{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 ^{2}}{1 + x ^{2}} d x - \int \frac{x}{1 + x ^{2}} d x

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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