integral of x/((3-x^2)^2)

解答

\int \frac{x}{( 3 - x ^{2} ) ^{2}} d x

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

求解步骤

\int \frac{x}{( 3 - x ^{2} ) ^{2}} d x

使用换元积分法

= \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

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

使用幂法则

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

u = 3 - x^{2}

代回

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

化简

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

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

解答补常数

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