integral of (x-sqrt(2x))/(x^3)

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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