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

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

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

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

Multipliziere aus

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

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

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

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

= - \frac{1}{x} - \frac{1}{2 x ^{2}}

Füge eine Konstante zur Lösung hinzu

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

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