integral of y^3e^{-y^2}

解答

\int y^{3} e^{- y^{2}} d y

\frac{1}{2} (- e^{- y^{2}} y^{2} - e^{- y^{2}}) + C

求解步骤

\int y^{3} e^{- y^{2}} d y

使用换元积分法

= \int \frac{e ^{u} u}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int e^{u} u d u

使用分布积分法

= \frac{1}{2} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= \frac{1}{2} (e^{u} u - e^{u})

u = - y^{2}

代回

= \frac{1}{2} (e^{- y^{2}} (- y^{2}) - e^{- y^{2}})

化简

= \frac{1}{2} (- e^{- y^{2}} y^{2} - e^{- y^{2}})

解答补常数

= \frac{1}{2} (- e^{- y^{2}} y^{2} - e^{- y^{2}}) + C

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