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

解

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

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

解答ステップ

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

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

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

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

部分積分を適用する

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

\int - 1 d x = - x

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

簡素化

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

定数を解答に追加する

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