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

解

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

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

解答ステップ

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

拡張

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

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

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

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

= arcsin (x) - 1 - x^{2}

定数を解答に追加する

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