x^4/(1-x^4) 1/(1-x^2)这两个函数的原函数（即积分）怎么算

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x^4/(1-x^4) 1/(1-x^2)这两个函数的原函数（即积分）怎么算
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x^4/(1-x^4) 1/(1-x^2)这两个函数的原函数（即积分）怎么算
x^4/(1-x^4) 1/(1-x^2)这两个函数的原函数（即积分）怎么算

x^4/(1-x^4) 1/(1-x^2)这两个函数的原函数（即积分）怎么算
先算后面一个
1/(1-x^2)dx=0.5(1/(1-x) dx+1/(1+x) dx)=0.5(ln|1+x|-ln|1-x|)=0.5ln|(1+x)/(1-x)|
再算前面一个
x^4/(1-x^4)dx
=(1/(1-x^4)-1)dx
=(0.5/(1-x^2)+0.5/(1+x^2)-1)dx
0.5/(1-x^2)的积分可以用到上面那个
0.5/(1+x^2)的积分有公式的=arctanx
所以答案是
0.25ln|(1+x)/(1-x)|+0.5arctanx-x
(注：答案中的常数项C未写)

  1. ∫x^4/(1-x^4)dx=∫[-1+1/(1-x^4)]dx

       =x+1/2∫[1/(1-x^2)+1/(1+x^2)]dx

       =x+1/2* arctanx+1/4∫[1/(1-x)+...

全部展开

  1. ∫x^4/(1-x^4)dx=∫[-1+1/(1-x^4)]dx

       =x+1/2∫[1/(1-x^2)+1/(1+x^2)]dx

       =x+1/2* arctanx+1/4∫[1/(1-x)+1/(1+x)]dx

       =x+1/2* arctanx+1/4ln|(1+x)/(1-x)|+C

2.   ∫1/(1-x^2)dx

      =1/2∫[1/(1-x)+1/(1+x)]dx

      =1/2ln|(1+x)/(1-x)|+C

收起

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