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Beliebt Rechnen >

integral of (x+1/x)^{3/2}((x^2-1)/(x^2))

Lösung

\int (x + \frac{1}{x})^{\frac{3}{2}} (\frac{x ^{2} - 1}{x ^{2}}) d x

Lösung

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

Schritte zur Lösung

\int (x + \frac{1}{x})^{\frac{3}{2}} \frac{x ^{2} - 1}{x ^{2}} d x

Wende U-Substitution an

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

Wende die Potenzregel an

= \frac{2}{5} u^{\frac{5}{2}}

Setze in

u = x + \frac{1}{x}

ein

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

Füge

x + \frac{1}{x}

zusammen:

\frac{x ^{2} + 1}{x}

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

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} (\frac{x ^{2} + 1}{x})^{\frac{5}{2}} + C

Graph

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