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受欢迎的微积分 >

integral of x^3e^{xy}

解答

\int x^{3} e^{x y}

解答

\frac{1}{y ^{4}} (y^{3} e^{y x} x^{3} - 3 (y^{2} e^{y x} x^{2} - 2 (y e^{y x} x - e^{y x}))) + C

求解步骤

\int x^{3} e^{x y} d x

使用换元积分法

= \int \frac{e ^{u} u ^{3}}{y ^{4}} d u

提出常数:

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

= \frac{1}{y ^{4}} \cdot \int e^{u} u^{3} d u

使用分布积分法

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

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

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

u = y x

代回

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

化简

\frac{1}{y ^{4}} ((y x)^{3} e^{y x} - 3 ((y x)^{2} e^{y x} - 2 (e^{y x} y x - e^{y x}))) : \frac{1}{y ^{4}} (y^{3} e^{y x} x^{3} - 3 (y^{2} e^{y x} x^{2} - 2 (y e^{y x} x - e^{y x})))

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

解答补常数

= \frac{1}{y ^{4}} (y^{3} e^{y x} x^{3} - 3 (y^{2} e^{y x} x^{2} - 2 (y e^{y x} x - e^{y x}))) + C

作图

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