integral of y/3 arcsin(y^2)

解

\int \frac{y}{3} arcsin (y^{2}) d y

\frac{1}{6} (y^{2} arcsin (y^{2}) + 1 - y^{4}) + C

解答ステップ

\int \frac{y}{3} arcsin (y^{2}) d y

定数を除く:

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

= \frac{1}{3} \cdot \int y arcsin (y^{2}) d y

部分積分を適用する

= \frac{1}{3} (\frac{1}{2} y^{2} arcsin (y^{2}) - \int \frac{y ^{3}}{1 - y ^{4}} d y)

\int \frac{y ^{3}}{1 - y ^{4}} d y = - \frac{1}{2} 1 - y^{4}

= \frac{1}{3} (\frac{1}{2} y^{2} arcsin (y^{2}) - (- \frac{1}{2} 1 - y^{4}))

簡素化

\frac{1}{3} (\frac{1}{2} y^{2} arcsin (y^{2}) - (- \frac{1}{2} 1 - y^{4})) : \frac{1}{6} (y^{2} arcsin (y^{2}) + 1 - y^{4})

= \frac{1}{6} (y^{2} arcsin (y^{2}) + 1 - y^{4})

定数を解答に追加する

= \frac{1}{6} (y^{2} arcsin (y^{2}) + 1 - y^{4}) + C