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受欢迎的微积分 >

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

解答

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

解答

- e^{- x^{2}} x^{2} - e^{- x^{2}} + C

求解步骤

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

提出常数:

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

= 2 \cdot \int x^{3} e^{- x^{2}} d x

使用换元积分法

= 2 \cdot \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

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

使用分布积分法

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

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

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

u = - x^{2}

代回

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

化简

2 \cdot \frac{1}{2} (e^{- x^{2}} (- x^{2}) - e^{- x^{2}}) : - e^{- x^{2}} x^{2} - e^{- x^{2}}

= - e^{- x^{2}} x^{2} - e^{- x^{2}}

解答补常数

= - e^{- x^{2}} x^{2} - e^{- x^{2}} + C

作图

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