integral of (sqrt(9-x^2))/(x^3)

Lösung

\int \frac{9 - x ^{2}}{x ^{3}} d x

- \frac{9 - x ^{2}}{2 x ^{2}} + \frac{1}{12} (ln \frac{1}{3} - x^{2} + 9 + 1 - ln \frac{1}{3} - x^{2} + 9 - 1) + C

Schritte zur Lösung

\int \frac{9 - x ^{2}}{x ^{3}} d x

Wende die partielle Integration an

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

\int \frac{1}{2 x - x ^{2} + 9} d x = - \frac{1}{12} (ln \frac{- x ^{2} + 9}{3} + 1 - ln \frac{- x ^{2} + 9}{3} - 1)

= - \frac{9 - x ^{2}}{2 x ^{2}} - (- \frac{1}{12} (ln \frac{- x ^{2} + 9}{3} + 1 - ln \frac{- x ^{2} + 9}{3} - 1))

Vereinfache

- \frac{9 - x ^{2}}{2 x ^{2}} - (- \frac{1}{12} (ln \frac{- x ^{2} + 9}{3} + 1 - ln \frac{- x ^{2} + 9}{3} - 1)) : - \frac{9 - x ^{2}}{2 x ^{2}} + \frac{1}{12} (ln \frac{1}{3} - x^{2} + 9 + 1 - ln \frac{1}{3} - x^{2} + 9 - 1)

= - \frac{9 - x ^{2}}{2 x ^{2}} + \frac{1}{12} (ln \frac{1}{3} - x^{2} + 9 + 1 - ln \frac{1}{3} - x^{2} + 9 - 1)

Füge eine Konstante zur Lösung hinzu

= - \frac{9 - x ^{2}}{2 x ^{2}} + \frac{1}{12} (ln \frac{1}{3} - x^{2} + 9 + 1 - ln \frac{1}{3} - x^{2} + 9 - 1) + C

\int \frac{1}{x ^{2} 16 - x ^{2}} d x

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