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受欢迎的微积分 >

integral of ((2y)/((x^2+y^2)^2))

解答

\int (\frac{2 y}{( x ^{2} + y ^{2} ) ^{2}}) d y

解答

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

求解步骤

\int \frac{2 y}{( x ^{2} + y ^{2} ) ^{2}} d y

提出常数:

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

= 2 \cdot \int \frac{y}{( x ^{2} + y ^{2} ) ^{2}} d y

使用换元积分法

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

使用幂法则

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

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

代回

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

化简

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

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

解答补常数

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

作图

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