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Beliebt Rechnen >

integral of ((x+x^{-2})^2)/(xsqrt(x))

Lösung

\int \frac{( x + x ^{- 2} ) ^{2}}{x x} d x

Lösung

\frac{2}{3} x^{\frac{3}{2}} - \frac{4}{3 x ^{\frac{3}{2}}} - \frac{2}{9 x ^{\frac{9}{2}}} + C

Schritte zur Lösung

\int \frac{( x + x ^{- 2} ) ^{2}}{x x} d x

Multipliziere aus

\frac{( x + x ^{- 2} ) ^{2}}{x x} : x + \frac{2}{x ^{2} x} + \frac{1}{x ^{5} x}

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

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

\int \frac{2}{x ^{2} x} d x = - \frac{4}{3 x ^{\frac{3}{2}}}

\int \frac{1}{x ^{5} x} d x = - \frac{2}{9 x ^{\frac{9}{2}}}

= \frac{2}{3} x^{\frac{3}{2}} - \frac{4}{3 x ^{\frac{3}{2}}} - \frac{2}{9 x ^{\frac{9}{2}}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} x^{\frac{3}{2}} - \frac{4}{3 x ^{\frac{3}{2}}} - \frac{2}{9 x ^{\frac{9}{2}}} + C

Graph

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