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

Lösung

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

\frac{1}{2} (x^{2} + 1 - ln x^{2} + 1) + 3 arctan (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

= \frac{1}{2} (x^{2} + 1 - ln x^{2} + 1) + 3 arctan (x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x^{2} + 1 - ln x^{2} + 1) + 3 arctan (x) + C

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