对(1-x^4)^-1积分?
来源:学生作业帮 编辑:搜搜做题作业网作业帮 分类:数学作业 时间:2024/07/28 18:42:03
对(1-x^4)^-1积分?
如题…
如题…
![对(1-x^4)^-1积分?](/uploads/image/z/18104529-57-9.jpg?t=%E5%AF%B9%281-x%5E4%29%5E-1%E7%A7%AF%E5%88%86%3F)
∫1/(1-x^4)dx
=∫1/(1+x²)(1-x²)dx
=[∫1/(1+x²)+1/(1-x²)dx]/2
=[∫1/(1+x²)dx]/2+[∫1/(1-x²)dx]/2
=[∫1/(1+x²)dx]/2+[∫1/(1+x)(1-x)dx]/2
=[∫1/(1+x²)dx]/2+[∫1/(1+x)+1/(1-x)dx]/4
=[∫1/(1+x²)dx]/2+[∫1/(1+x)dx]/4-[∫1/(1-x)d(-x)]/4
=arctanx/2+ln(1+x)/4-ln(1-x)/4+C
=∫1/(1+x²)(1-x²)dx
=[∫1/(1+x²)+1/(1-x²)dx]/2
=[∫1/(1+x²)dx]/2+[∫1/(1-x²)dx]/2
=[∫1/(1+x²)dx]/2+[∫1/(1+x)(1-x)dx]/2
=[∫1/(1+x²)dx]/2+[∫1/(1+x)+1/(1-x)dx]/4
=[∫1/(1+x²)dx]/2+[∫1/(1+x)dx]/4-[∫1/(1-x)d(-x)]/4
=arctanx/2+ln(1+x)/4-ln(1-x)/4+C