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Beliebt Rechnen >

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

Lösung

\int x^{2} arcsin (1 - x^{2}) d x

Lösung

- \frac{1}{3} arcsin (1 - x^{2}) (1 - x^{2})^{\frac{3}{2}} + \frac{1}{9} (- 3 x + x^{3}) + C

Schritte zur Lösung

\int x^{2} arcsin (1 - x^{2}) d x

Wende U-Substitution an

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Wende die partielle Integration an

= - (\frac{1}{3} u sin^{3} (u) - \int \frac{1}{3} sin^{3} (u) d u)

\int \frac{1}{3} sin^{3} (u) d u = \frac{1}{3} (- cos (u) + \frac{cos ^{3} ( u )}{3})

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

Setze in

u = arcsin (1 - x^{2})

ein

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

Vereinfache

- (\frac{1}{3} arcsin (1 - x^{2}) sin^{3} (arcsin (1 - x^{2})) - \frac{1}{3} (- cos (arcsin (1 - x^{2})) + \frac{cos ^{3} ( arcsin ( 1 - x ^{2} ) )}{3})) : - \frac{1}{3} arcsin (1 - x^{2}) (1 - x^{2})^{\frac{3}{2}} + \frac{1}{9} (- 3 x + x^{3})

= - \frac{1}{3} arcsin (1 - x^{2}) (1 - x^{2})^{\frac{3}{2}} + \frac{1}{9} (- 3 x + x^{3})

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} arcsin (1 - x^{2}) (1 - x^{2})^{\frac{3}{2}} + \frac{1}{9} (- 3 x + x^{3}) + C

Graph

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