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Beliebt Rechnen >

integral of (arcsin(1/x))/(x^5)

Lösung

\int \frac{arcsin ( \frac{1}{x} )}{x ^{5}} d x

Lösung

- \frac{arcsin ( \frac{1}{x} )}{4 x ^{4}} - \frac{x ^{2} - 1}{16 x ^{4}} + \frac{3}{32} arcsin (\frac{1}{x}) - \frac{3}{64} sin (2 arcsin (\frac{1}{x})) + C

Schritte zur Lösung

\int \frac{arcsin ( \frac{1}{x} )}{x ^{5}} d x

Wende U-Substitution an

= \int - u^{3} arcsin (u) d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int u^{3} arcsin (u) d u

Wende die partielle Integration an

= - (\frac{1}{4} u^{4} arcsin (u) - \int \frac{u ^{4}}{4 1 - u ^{2}} d u)

\int \frac{u ^{4}}{4 1 - u ^{2}} d u = \frac{1}{4} (- \frac{1}{4} u^{3} 1 - u^{2} + \frac{3}{8} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u))))

= - (\frac{1}{4} u^{4} arcsin (u) - \frac{1}{4} (- \frac{1}{4} u^{3} 1 - u^{2} + \frac{3}{8} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u)))))

Setze in

u = \frac{1}{x}

ein

= - \frac{1}{4} (\frac{1}{x})^{4} arcsin (\frac{1}{x}) - \frac{1}{4} - \frac{1}{4} (\frac{1}{x})^{3} 1 - (\frac{1}{x})^{2} + \frac{3}{8} (arcsin (\frac{1}{x}) - \frac{1}{2} sin (2 arcsin (\frac{1}{x})))

Vereinfache

- \frac{1}{4} (\frac{1}{x})^{4} arcsin (\frac{1}{x}) - \frac{1}{4} - \frac{1}{4} (\frac{1}{x})^{3} 1 - (\frac{1}{x})^{2} + \frac{3}{8} (arcsin (\frac{1}{x}) - \frac{1}{2} sin (2 arcsin (\frac{1}{x}))) : - \frac{arcsin ( \frac{1}{x} )}{4 x ^{4}} - \frac{x ^{2} - 1}{16 x ^{4}} + \frac{3}{32} arcsin (\frac{1}{x}) - \frac{3}{64} sin (2 arcsin (\frac{1}{x}))

= - \frac{arcsin ( \frac{1}{x} )}{4 x ^{4}} - \frac{x ^{2} - 1}{16 x ^{4}} + \frac{3}{32} arcsin (\frac{1}{x}) - \frac{3}{64} sin (2 arcsin (\frac{1}{x}))

Füge eine Konstante zur Lösung hinzu

= - \frac{arcsin ( \frac{1}{x} )}{4 x ^{4}} - \frac{x ^{2} - 1}{16 x ^{4}} + \frac{3}{32} arcsin (\frac{1}{x}) - \frac{3}{64} sin (2 arcsin (\frac{1}{x})) + C

Graph

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