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

Lösung

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

\frac{1}{4} x^{3} x^{2} - 1 - \frac{1}{8} x x^{2} - 1 - \frac{1}{8} ln x^{2} - 1 + x + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{1}{3} x^{3} x^{2} - 1 - \int \frac{x ^{4}}{3 x ^{2} - 1} d x

\int \frac{x ^{4}}{3 x ^{2} - 1} d x = \frac{1}{12} x^{3} x^{2} - 1 + \frac{1}{8} (x x^{2} - 1 + ln x^{2} - 1 + x)

= \frac{1}{3} x^{3} x^{2} - 1 - (\frac{1}{12} x^{3} x^{2} - 1 + \frac{1}{8} (x x^{2} - 1 + ln x^{2} - 1 + x))

Vereinfache

\frac{1}{3} x^{3} x^{2} - 1 - (\frac{1}{12} x^{3} x^{2} - 1 + \frac{1}{8} (x x^{2} - 1 + ln x^{2} - 1 + x)) : \frac{1}{4} x^{3} x^{2} - 1 - \frac{1}{8} x x^{2} - 1 - \frac{1}{8} ln x^{2} - 1 + x

= \frac{1}{4} x^{3} x^{2} - 1 - \frac{1}{8} x x^{2} - 1 - \frac{1}{8} ln x^{2} - 1 + x

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} x^{3} x^{2} - 1 - \frac{1}{8} x x^{2} - 1 - \frac{1}{8} ln x^{2} - 1 + x + C