integral of (x-3)/(sqrt(1-2x-x^2))

解

\int \frac{x - 3}{1 - 2 x - x ^{2}} d x

- - x^{2} - 2 x + 1 - 4 arcsin (\frac{1}{2} (x + 1)) + C

解答ステップ

\int \frac{x - 3}{1 - 2 x - x ^{2}} d x

拡張

\frac{x - 3}{1 - 2 x - x ^{2}} : \frac{x}{1 - 2 x - x ^{2}} - \frac{3}{1 - 2 x - x ^{2}}

= \int \frac{x}{1 - 2 x - x ^{2}} - \frac{3}{1 - 2 x - 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 \frac{x}{1 - 2 x - x ^{2}} d x - \int \frac{3}{1 - 2 x - x ^{2}} d x

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

\int \frac{3}{1 - 2 x - x ^{2}} d x = 3 arcsin (\frac{1}{2} (x + 1))

= - - x^{2} - 2 x + 1 - arcsin (\frac{1}{2} (x + 1)) - 3 arcsin (\frac{1}{2} (x + 1))

簡素化

= - - x^{2} - 2 x + 1 - 4 arcsin (\frac{1}{2} (x + 1))

定数を解答に追加する

= - - x^{2} - 2 x + 1 - 4 arcsin (\frac{1}{2} (x + 1)) + C