不定积分∫(e∧x-1)/(e∧x+1)等于,麻烦给出过程.
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/04 21:12:52
![不定积分∫(e∧x-1)/(e∧x+1)等于,麻烦给出过程.](/uploads/image/z/14058326-38-6.jpg?t=%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%E2%88%AB%28e%E2%88%A7x-1%29%2F%28e%E2%88%A7x%2B1%29%E7%AD%89%E4%BA%8E%2C%E9%BA%BB%E7%83%A6%E7%BB%99%E5%87%BA%E8%BF%87%E7%A8%8B.)
不定积分∫(e∧x-1)/(e∧x+1)等于,麻烦给出过程.
不定积分∫(e∧x-1)/(e∧x+1)等于,麻烦给出过程.
不定积分∫(e∧x-1)/(e∧x+1)等于,麻烦给出过程.
∫ (e^x - 1)/(e^x + 1) dx
= ∫ (e^x + 1 - 2)/(e^x + 1) dx
= ∫ dx - 2∫ dx/(e^x + 1)
= x - 2∫ e^x/[e^x(e^x + 1)] dx
= x - 2∫ (e^x + 1 - e^x)/[e^x(e^x + 1)] d(e^x)
= x - 2∫ 1/e^x d(e^x)...
全部展开
∫ (e^x - 1)/(e^x + 1) dx
= ∫ (e^x + 1 - 2)/(e^x + 1) dx
= ∫ dx - 2∫ dx/(e^x + 1)
= x - 2∫ e^x/[e^x(e^x + 1)] dx
= x - 2∫ (e^x + 1 - e^x)/[e^x(e^x + 1)] d(e^x)
= x - 2∫ 1/e^x d(e^x) + 2∫ 1/(e^x + 1) d(e^x + 1)
= x - 2ln(e^x) + 2ln(1 + e^x) + C
= x - 2ln[e^x/(1 + e^x)] + C
= x + 2ln[1 + e^(- x)] + C
收起
原式=∫ [e^x/(e^x+1)-1/(e^x+1)]dx
=∫d(e^x)/(e^x+1)- ∫ d(e^x)/[e^x(e^x+1)]
=ln(e^x+1)-∫d(e^x)*[1/e^x-1/(e^x+1)]
=ln(e^x+1)-lne^x+ln(e^x+1)+C
=2ln(e^x+1)-x+C