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

\int x (3 x + 1)^{2} d x

x \to 1 lim (ln (\frac{1}{e} x))

\frac{d ^{2}}{d x ^{2}} (5 (x^{2} + 8)^{4})

\int (x + 4) (x + 3)^{\frac{1}{4}} d x

(cos (arcsin (x)))^{^{'}}