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

解

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

- \frac{x}{arcsin ( x )} + Ci (arcsin (x)) + C

解答ステップ

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

u置換積分法を適用する

= \int \frac{sin ( u )}{u ^{2}} d u

部分積分を適用する

= - \frac{sin ( u )}{u} - \int - \frac{cos ( u )}{u} d u

\int - \frac{cos ( u )}{u} d u = - Ci (u)

= - \frac{sin ( u )}{u} - (- Ci (u))

代用を戻す

u = arcsin (x)

= - \frac{sin ( arcsin ( x ) )}{arcsin ( x )} - (- Ci (arcsin (x)))

簡素化

- \frac{sin ( arcsin ( x ) )}{arcsin ( x )} - (- Ci (arcsin (x))) : - \frac{x}{arcsin ( x )} + Ci (arcsin (x))

= - \frac{x}{arcsin ( x )} + Ci (arcsin (x))

定数を解答に追加する

= - \frac{x}{arcsin ( x )} + Ci (arcsin (x)) + C