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

Lösung

\int (x - 25 - x^{2}) d x

\frac{x ^{2}}{2} - \frac{25}{2} (arcsin (\frac{1}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} x))) + C

Schritte zur Lösung

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

Wende die Summenregel an:

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

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

\int x d x = \frac{x ^{2}}{2}

\int 25 - x^{2} d x = \frac{25}{2} (arcsin (\frac{1}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} x)))

= \frac{x ^{2}}{2} - \frac{25}{2} (arcsin (\frac{1}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} x)))

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - \frac{25}{2} (arcsin (\frac{1}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} x))) + C

x \to 4 lim (sin (\frac{2}{x} - \frac{1}{2}))

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