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

Lösung

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

\frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{x ^{2}}{x} d x = \frac{2}{5} x^{\frac{5}{2}}

= \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

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