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

Lösung

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

- \frac{1}{6} x^{3} + \frac{x ^{3}}{2} + \frac{1}{2} x^{2} x^{2} - 1 - \frac{1}{6} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{2} x^{2} - 1 + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int \frac{1}{2} (x^{2} + x x^{2} - 1 - ln x + x^{2} - 1) d x = \frac{1}{6} (x^{3} + (x^{2} - 1)^{\frac{3}{2}} - 3 (x ln x + x^{2} - 1 - x^{2} - 1))

= x (\frac{x ^{2}}{2} + \frac{1}{2} x x^{2} - 1 - \frac{1}{2} ln x + x^{2} - 1) - \frac{1}{6} (x^{3} + (x^{2} - 1)^{\frac{3}{2}} - 3 (x ln x + x^{2} - 1 - x^{2} - 1))

Vereinfache

x (\frac{x ^{2}}{2} + \frac{1}{2} x x^{2} - 1 - \frac{1}{2} ln x + x^{2} - 1) - \frac{1}{6} (x^{3} + (x^{2} - 1)^{\frac{3}{2}} - 3 (x ln x + x^{2} - 1 - x^{2} - 1)) : - \frac{1}{6} x^{3} + \frac{x ^{3}}{2} + \frac{1}{2} x^{2} x^{2} - 1 - \frac{1}{6} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{2} x^{2} - 1

= - \frac{1}{6} x^{3} + \frac{x ^{3}}{2} + \frac{1}{2} x^{2} x^{2} - 1 - \frac{1}{6} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{2} x^{2} - 1

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{6} x^{3} + \frac{x ^{3}}{2} + \frac{1}{2} x^{2} x^{2} - 1 - \frac{1}{6} (x^{2} - 1)^{\frac{3}{2}} - \frac{1}{2} x^{2} - 1 + C

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