integral of (x^4)/((sqrt(1-x^2))^3)

Lösung

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

\frac{x ^{5}}{1 - x ^{2}} + x^{3} 1 - x^{2} - \frac{3}{2} arcsin (x) + \frac{3}{4} sin (2 arcsin (x)) + C

Schritte zur Lösung

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

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

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

Wende die partielle Integration an

= \frac{x ^{5}}{1 - x ^{2}} - \int \frac{4 x ^{4}}{1 - x ^{2}} d x

\int \frac{4 x ^{4}}{1 - x ^{2}} d x = - x^{3} 1 - x^{2} + \frac{3}{2} arcsin (x) - \frac{3}{4} sin (2 arcsin (x))

= \frac{x ^{5}}{1 - x ^{2}} - (- x^{3} 1 - x^{2} + \frac{3}{2} arcsin (x) - \frac{3}{4} sin (2 arcsin (x)))

Vereinfache

= \frac{x ^{5}}{1 - x ^{2}} + x^{3} 1 - x^{2} - \frac{3}{2} arcsin (x) + \frac{3}{4} sin (2 arcsin (x))

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{5}}{1 - x ^{2}} + x^{3} 1 - x^{2} - \frac{3}{2} arcsin (x) + \frac{3}{4} sin (2 arcsin (x)) + C

d er i v a t i v e x^{2} ln (3 x)

\int x sin (x y) d y

x \to - 1 lim (2 x - 3)

x \to \infty lim ((1 + \frac{8}{x})^{\frac{x}{9}})

x \to 0 lim (ln (\frac{1}{x}))