integral of (x-3)/(\sqrt[3]{x^5)}

Lösung

\int \frac{x - 3}{3 x ^{5}} d x

3 3 x + \frac{9}{2 x ^{\frac{2}{3}}} + C

Schritte zur Lösung

\int \frac{x - 3}{3 x ^{5}} d x

3 x^{5} = x^{\frac{5}{3}},

angenommen

x \geq 0

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

Multipliziere aus

\frac{x - 3}{x ^{\frac{5}{3}}} : \frac{1}{x ^{\frac{2}{3}}} - \frac{3}{x ^{\frac{5}{3}}}

= \int \frac{1}{x ^{\frac{2}{3}}} - \frac{3}{x ^{\frac{5}{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 ^{\frac{2}{3}}} d x - \int \frac{3}{x ^{\frac{5}{3}}} d x

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

\int \frac{3}{x ^{\frac{5}{3}}} d x = - \frac{9}{2 x ^{\frac{2}{3}}}

= 3 x^{\frac{1}{3}} - (- \frac{9}{2 x ^{\frac{2}{3}}})

Vereinfache

3 x^{\frac{1}{3}} - (- \frac{9}{2 x ^{\frac{2}{3}}}) : 3 3 x + \frac{9}{2 x ^{\frac{2}{3}}}

= 3 3 x + \frac{9}{2 x ^{\frac{2}{3}}}

Füge eine Konstante zur Lösung hinzu

= 3 3 x + \frac{9}{2 x ^{\frac{2}{3}}} + C

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