integral of 2xarcsin(x)

解

\int 2 x arcsin (x) d x

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

解答ステップ

\int 2 x arcsin (x) d x

定数を除く:

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

= 2 \cdot \int x arcsin (x) d x

部分積分を適用する

= 2 (\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)))

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

簡素化

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

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

定数を解答に追加する

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