integral of 1/(x^3)sqrt(1-1/(x^2))

Lösung

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

- \frac{x ^{2} - 1}{3 x ^{3}} + \frac{x ^{2} - 1}{3 x} + C

Schritte zur Lösung

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

Vereinfache

\frac{1}{x ^{3}} 1 - \frac{1}{x ^{2}} : \frac{x ^{2} - 1}{x ^{3} x ^{2}}

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

x^{2} = (x),

angenommen

x \geq 0

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

x^{3} x = x^{4}

= \int \frac{x ^{2} - 1}{x ^{4}} d x

Wende die partielle Integration an

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

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

= - \frac{x ^{2} - 1}{3 x ^{3}} - (- \frac{x ^{2} - 1}{3 x})

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= - \frac{x ^{2} - 1}{3 x ^{3}} + \frac{x ^{2} - 1}{3 x} + C

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