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

Lösung

\int 1 - x^{2} - y^{2} d y

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

Schritte zur Lösung

\int 1 - x^{2} - y^{2} d y

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

= 1 - x^{2} \cdot \int cos (u) - x^{2} + x^{2} sin^{2} (u) + cos^{2} (u) d u

Wende U-Substitution an

= 1 - x^{2} \cdot \int - \frac{v ^{2} - x ^{2} + 1}{1 - v ^{2}} d v

Entferne die Konstante:

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

= 1 - x^{2} (- - x^{2} + 1 \cdot \int \frac{v ^{2}}{1 - v ^{2}} d v)

Trigonometrische Substitution anwenden

= 1 - x^{2} (- - x^{2} + 1 \cdot \int sin^{2} (w) d w)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 1 - x^{2} (- - x^{2} + 1 \cdot \int \frac{1 - cos ( 2 w )}{2} d w)

Entferne die Konstante:

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

= 1 - x^{2} (- - x^{2} + 1 \frac{1}{2} \cdot \int 1 - cos (2 w) d w)

Wende die Summenregel an:

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

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

\int 1 d w = w

\int cos (2 w) d w = \frac{1}{2} sin (2 w)

= 1 - x^{2} (- - x^{2} + 1 \frac{1}{2} (w - \frac{1}{2} sin (2 w)))

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{1}{t ^{2} - 8 t + 41} d t

\frac{d}{d x} (8 x (x^{5} + 1)^{3})

\frac{\partial}{\partial x} (4 + x + x^{2} y)

x \to 0 lim (\frac{4}{x})

l a pl a ce t r an s f or m e^{- 3 s}