Question

请问如下不定积分如何求解

x2 除以 （ x4 + 1 ）

困扰了我好久了，求大神相助！！！

Answer 1

以下是用Matlab计算出来的

不过可以发现，它这个解法可以视为将分母因式分解(在复数范围内)，再把分式拆成更简单的分式继续积分的。受其启发，手动计算的话，也可以有类似解法。

Answer 2

x^4+1 = x^4+2x^2+1 - 2x^2
= (x^2+1)^2 - 2x^2 = (x^2+√2x+1)(x^2-√2x+1)
x^2/(x^4+1) = (√2/4)[x/(x^2-√2x+1) - x/(x^2+√2x+1)]
∫x^2dx/(x^4+1) = (√2/4)∫xdx/(x^2-√2x+1) - (√2/4)∫xdx/(x^2+√2x+1)
= (√2/8)∫(2x-√2+√2)dx/(x^2-√2x+1) - (√2/8)∫(2x+√2-√2)dx/(x^2+√2x+1)
= (√2/8)∫d(x^2-√2x+1)/(x^2-√2x+1) + (1/4)∫d(x-√2/2)/[(x-√2/2)^2+1/2]
-(√2/8)∫d(x^2+√2x+1)/(x^2+√2x+1) + (1/4)∫d(x+√2/2)/[(x+√2/2)^2+1/2]
= (√2/8)ln[(x^2-√2x+1)/(x^2+√2x+1)] + (√2/4)arctan(√2x-1)
+ (√2/4)arctan(√2x+1) + C

Answer 3

如图所示：