integral of (x^3-x)/((x^4-2x^2+3)^2)

解答

\int \frac{x ^{3} - x}{( x ^{4} - 2 x ^{2} + 3 ) ^{2}} d x

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

求解步骤

\int \frac{x ^{3} - x}{( x ^{4} - 2 x ^{2} + 3 ) ^{2}} d x

使用换元积分法

= \int \frac{1}{4 u ^{2}} d u

提出常数:

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

= \frac{1}{4} \cdot \int \frac{1}{u ^{2}} d u

使用幂法则

= \frac{1}{4} (- \frac{1}{u})

u = x^{4} - 2 x^{2} + 3

代回

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

化简

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

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

解答补常数

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

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