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

Lösung

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

ln ∣ x ∣ + arctan (x) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + arctan (x)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + arctan (x) + C

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