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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{4} x^{3} 1 - x^{2} + \frac{3}{8} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))) + C

Schritte zur Lösung

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

Vereinfache

x^{4} \frac{1}{1 - x ^{2}} : \frac{x ^{4}}{1 - x ^{2}}

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

Trigonometrische Substitution anwenden

= \int sin^{4} (u) d u

Wende integrale Reduktion an

= - \frac{cos ( u ) sin ^{3} ( u )}{4} + \frac{3}{4} \cdot \int sin^{2} (u) d u

\int sin^{2} (u) d u = \frac{1}{2} (u - \frac{1}{2} sin (2 u))

= - \frac{cos ( u ) sin ^{3} ( u )}{4} + \frac{3}{4} \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

Setze in

u = arcsin (x)

ein

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

Vereinfache

- \frac{cos ( arcsin ( x ) ) sin ^{3} ( arcsin ( x ) )}{4} + \frac{3}{4} \cdot \frac{1}{2} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))) : - \frac{1}{4} x^{3} 1 - x^{2} + \frac{3}{8} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))

= - \frac{1}{4} x^{3} 1 - x^{2} + \frac{3}{8} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} x^{3} 1 - x^{2} + \frac{3}{8} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))) + C

Graph

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