integral of x^2sqrt(81-4x^2)

Lösung

\int x^{2} 81 - 4 x^{2} d x

\frac{6561}{64} (arcsin (\frac{2}{9} x) - \frac{1}{4} sin (4 arcsin (\frac{2}{9} x))) + C

Schritte zur Lösung

\int x^{2} 81 - 4 x^{2} d x

Trigonometrische Substitution anwenden

= \int \frac{6561}{8} sin^{2} (u) cos^{2} (u) d u

Entferne die Konstante:

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

= \frac{6561}{8} \cdot \int sin^{2} (u) cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{6561}{8} \cdot \int \frac{1 - cos ( 4 u )}{8} d u

Entferne die Konstante:

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

= \frac{6561}{8} \cdot \frac{1}{8} \cdot \int 1 - cos (4 u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{6561}{8} \cdot \frac{1}{8} (\int 1 d u - \int cos (4 u) d u)

\int 1 d u = u

\int cos (4 u) d u = \frac{1}{4} sin (4 u)

= \frac{6561}{8} \cdot \frac{1}{8} (u - \frac{1}{4} sin (4 u))

Setze in

u = arcsin (\frac{2}{9} x)

ein

= \frac{6561}{8} \cdot \frac{1}{8} (arcsin (\frac{2}{9} x) - \frac{1}{4} sin (4 arcsin (\frac{2}{9} x)))

Vereinfache

\frac{6561}{8} \cdot \frac{1}{8} (arcsin (\frac{2}{9} x) - \frac{1}{4} sin (4 arcsin (\frac{2}{9} x))) : \frac{6561}{64} (arcsin (\frac{2}{9} x) - \frac{1}{4} sin (4 arcsin (\frac{2}{9} x)))

= \frac{6561}{64} (arcsin (\frac{2}{9} x) - \frac{1}{4} sin (4 arcsin (\frac{2}{9} x)))

Füge eine Konstante zur Lösung hinzu

= \frac{6561}{64} (arcsin (\frac{2}{9} x) - \frac{1}{4} sin (4 arcsin (\frac{2}{9} x))) + C

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