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Beliebt Rechnen >

integral of 2x(arcsin(x))^2

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

= 2 \cdot \int x (arcsin (x))^{2} d x

Vereinfache

= 2 \cdot \int x arcsin^{2} (x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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