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受欢迎的微积分 >

integral of x/((2x^2-1)^2)

解答

\int \frac{x}{( 2 x ^{2} - 1 ) ^{2}} d x

解答

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

求解步骤

\int \frac{x}{( 2 x ^{2} - 1 ) ^{2}} d x

因式分解

2 x^{2} - 1 : - (- 2 x^{2} + 1)

= \int \frac{x}{( - ( - 2 x ^{2} + 1 ) ) ^{2}} d x

(- (- 2 x^{2} + 1))^{2} = (- 2 x^{2} + 1)^{2}

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

代回

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

化简

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

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

解答补常数

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

作图

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