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Beliebt Rechnen >

integral of sqrt(25-x^2-y^2)

Lösung

\int 25 - x^{2} - y^{2} d x

Lösung

\frac{1}{2} (25 - y^{2}) (arcsin (\frac{x}{25 - y ^{2}}) + \frac{1}{2} sin (2 arcsin (\frac{x}{25 - y ^{2}}))) + C

Schritte zur Lösung

\int 25 - x^{2} - y^{2} d x

Trigonometrische Substitution anwenden

= \int cos^{2} (u) (25 - y^{2}) d u

Entferne die Konstante:

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

= (25 - y^{2}) \cdot \int cos^{2} (u) d u

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

= (25 - y^{2}) \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Setze in

u = arcsin (\frac{1}{25 - y ^{2}} x)

ein

= (25 - y^{2}) \frac{1}{2} (arcsin (\frac{1}{25 - y ^{2}} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{25 - y ^{2}} x)))

Vereinfache

(25 - y^{2}) \frac{1}{2} (arcsin (\frac{1}{25 - y ^{2}} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{25 - y ^{2}} x))) : \frac{1}{2} (25 - y^{2}) (arcsin (\frac{x}{25 - y ^{2}}) + \frac{1}{2} sin (2 arcsin (\frac{x}{25 - y ^{2}})))

= \frac{1}{2} (25 - y^{2}) (arcsin (\frac{x}{25 - y ^{2}}) + \frac{1}{2} sin (2 arcsin (\frac{x}{25 - y ^{2}})))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (25 - y^{2}) (arcsin (\frac{x}{25 - y ^{2}}) + \frac{1}{2} sin (2 arcsin (\frac{x}{25 - y ^{2}}))) + C

Graph

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