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

Lösung

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

\frac{x ^{2}}{2} + \frac{1}{2} ln x^{2} + 2 x + C

Schritte zur Lösung

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

Lange Division

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} + \frac{1}{2} ln x^{2} + 2 x + C