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受欢迎的微积分 >

integral of xye^{x^2+y^2}

解答

\int x y e^{x^{2} + y^{2}} d x

解答

\frac{1}{2} y e^{y^{2} + x^{2}} + C

求解步骤

\int x y e^{x^{2} + y^{2}} d x

提出常数:

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

= y \cdot \int e^{y^{2} + x^{2}} x d x

使用换元积分法

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

提出常数:

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

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

化简

e^{y^{2} + u} : e^{y^{2}} e^{u}

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

提出常数:

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

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

使用积分常见定则:

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

= y \frac{1}{2} e^{y^{2}} e^{u}

u = x^{2}

代回

= y \frac{1}{2} e^{y^{2}} e^{x^{2}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

e^{y^{2}} e^{x^{2}} = e^{y^{2} + x^{2}}

= \frac{1}{2} y e^{y^{2} + x^{2}}

解答补常数

= \frac{1}{2} y e^{y^{2} + x^{2}} + C

作图

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