integral of (sqrt(4-x^2))/(x^3)

Lösung

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

- \frac{4 - x ^{2}}{2 x ^{2}} + \frac{1}{8} (ln \frac{1}{2} - x^{2} + 4 + 1 - ln \frac{1}{2} - x^{2} + 4 - 1) + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int \frac{1}{2 x - x ^{2} + 4} d x = - \frac{1}{8} (ln \frac{- x ^{2} + 4}{2} + 1 - ln \frac{- x ^{2} + 4}{2} - 1)

= - \frac{4 - x ^{2}}{2 x ^{2}} - (- \frac{1}{8} (ln \frac{- x ^{2} + 4}{2} + 1 - ln \frac{- x ^{2} + 4}{2} - 1))

Vereinfache

- \frac{4 - x ^{2}}{2 x ^{2}} - (- \frac{1}{8} (ln \frac{- x ^{2} + 4}{2} + 1 - ln \frac{- x ^{2} + 4}{2} - 1)) : - \frac{4 - x ^{2}}{2 x ^{2}} + \frac{1}{8} (ln \frac{1}{2} - x^{2} + 4 + 1 - ln \frac{1}{2} - x^{2} + 4 - 1)

= - \frac{4 - x ^{2}}{2 x ^{2}} + \frac{1}{8} (ln \frac{1}{2} - x^{2} + 4 + 1 - ln \frac{1}{2} - x^{2} + 4 - 1)

Füge eine Konstante zur Lösung hinzu

= - \frac{4 - x ^{2}}{2 x ^{2}} + \frac{1}{8} (ln \frac{1}{2} - x^{2} + 4 + 1 - ln \frac{1}{2} - x^{2} + 4 - 1) + C

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