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受欢迎的微积分 >

integral of 5sqrt(x)e^{sqrt(x)}

解答

\int 5 x e^{x} d x

解答

10 (e^{x} x - 2 (e^{x} x - e^{x})) + C

求解步骤

\int 5 x e^{x} d x

提出常数:

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

= 5 \cdot \int x e^{x} d x

使用换元积分法

= 5 \cdot \int 2 e^{u} u^{2} d u

提出常数:

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

= 5 \cdot 2 \cdot \int e^{u} u^{2} d u

使用分布积分法

= 5 \cdot 2 (u^{2} e^{u} - \int 2 u e^{u} d u)

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

= 5 \cdot 2 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

u = x

代回

= 5 \cdot 2 ((x)^{2} e^{x} - 2 (e^{x} x - e^{x}))

化简

5 \cdot 2 ((x)^{2} e^{x} - 2 (e^{x} x - e^{x})) : 10 (e^{x} x - 2 (e^{x} x - e^{x}))

= 10 (e^{x} x - 2 (e^{x} x - e^{x}))

解答补常数

= 10 (e^{x} x - 2 (e^{x} x - e^{x})) + C

作图

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