integral of sqrt(r^2-x^2)

Lösung

\int r^{2} - x^{2} d x

\frac{1}{2} r^{2} (arcsin (\frac{x}{r}) + \frac{1}{2} sin (2 arcsin (\frac{x}{r}))) + C

Schritte zur Lösung

\int r^{2} - x^{2} d x

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

= r^{2} \cdot \int cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Entferne die Konstante:

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

= r^{2} \frac{1}{2} \cdot \int 1 + cos (2 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

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

\int 1 d u = u

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

= r^{2} \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Setze in

u = arcsin (\frac{1}{r} x)

ein

= r^{2} \frac{1}{2} (arcsin (\frac{1}{r} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{r} x)))

Vereinfache

r^{2} \frac{1}{2} (arcsin (\frac{1}{r} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{r} x))) : \frac{1}{2} r^{2} (arcsin (\frac{x}{r}) + \frac{1}{2} sin (2 arcsin (\frac{x}{r})))

= \frac{1}{2} r^{2} (arcsin (\frac{x}{r}) + \frac{1}{2} sin (2 arcsin (\frac{x}{r})))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} r^{2} (arcsin (\frac{x}{r}) + \frac{1}{2} sin (2 arcsin (\frac{x}{r}))) + C

\frac{d y}{d x} = 6 - x

t an g e n t f (x) = 2 x^{2} - 10 x + 9, (2, - 3)

(x^{2} + y^{2}) d x + 2 x y d y = 0

n = 1 \sum \infty \frac{5 n}{n ^{2} + 2}

\frac{d}{d x} (x^{3} - \frac{1}{x ^{7}})