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

Lösung

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

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

Schritte zur Lösung

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

Lange Division

\frac{x ^{3} + x + 1}{x ^{2} + 1} : x + \frac{1}{x ^{2} + 1}

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

\int x d x = \frac{x ^{2}}{2}

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

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

Füge eine Konstante zur Lösung hinzu

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

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