integral of (x^2-x^5+sqrt(x))/(x^5)

Lösung

\int \frac{x ^{2} - x ^{5} + x}{x ^{5}} d x

- \frac{1}{2 x ^{2}} - x - \frac{2}{7 x ^{\frac{7}{2}}} + C

Schritte zur Lösung

\int \frac{x ^{2} - x ^{5} + x}{x ^{5}} d x

Multipliziere aus

\frac{x ^{2} - x ^{5} + x}{x ^{5}} : \frac{1}{x ^{3}} - 1 + \frac{1}{x ^{\frac{9}{2}}}

= \int \frac{1}{x ^{3}} - 1 + \frac{1}{x ^{\frac{9}{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 \frac{1}{x ^{3}} d x - \int 1 d x + \int \frac{1}{x ^{\frac{9}{2}}} d x

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

\int 1 d x = x

\int \frac{1}{x ^{\frac{9}{2}}} d x = - \frac{2}{7 x ^{\frac{7}{2}}}

= - \frac{1}{2 x ^{2}} - x - \frac{2}{7 x ^{\frac{7}{2}}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2 x ^{2}} - x - \frac{2}{7 x ^{\frac{7}{2}}} + C

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