integral of (sqrt(x)-x)/(\sqrt[3]{x^2)}

Lösung

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

\frac{6}{5} x^{\frac{5}{6}} - \frac{3}{4} x^{\frac{4}{3}} + C

Schritte zur Lösung

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

3 x^{2} = x^{\frac{2}{3}},

angenommen

x \geq 0

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

Multipliziere aus

\frac{x - x}{x ^{\frac{2}{3}}} : \frac{1}{6 x} - 3 x

= \int \frac{1}{6 x} - 3 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 \frac{1}{6 x} d x - \int 3 x d x

\int \frac{1}{6 x} d x = \frac{6}{5} x^{\frac{5}{6}}

\int 3 x d x = \frac{3}{4} x^{\frac{4}{3}}

= \frac{6}{5} x^{\frac{5}{6}} - \frac{3}{4} x^{\frac{4}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{6}{5} x^{\frac{5}{6}} - \frac{3}{4} x^{\frac{4}{3}} + C

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