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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

\int 1 d x = x

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

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

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

Füge eine Konstante zur Lösung hinzu

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