integral of xarcsin(x))

解

\int x d (arcsin (x)) d x

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

解答ステップ

\int x d arcsin (x) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= d \cdot \int x arcsin (x) d x

部分積分を適用する

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

\int \frac{x ^{2}}{2 1 - x ^{2}} d x = \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))

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

定数を解答に追加する

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