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

解

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

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

解答ステップ

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

総和規則を適用する:

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

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

\int 1 d x = x

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

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

定数を解答に追加する

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