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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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