高数简单微分计算∫[(2x+1)/(x^2+2x+2])dx
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/23 18:37:28
![高数简单微分计算∫[(2x+1)/(x^2+2x+2])dx](/uploads/image/z/6083179-43-9.jpg?t=%E9%AB%98%E6%95%B0%E7%AE%80%E5%8D%95%E5%BE%AE%E5%88%86%E8%AE%A1%E7%AE%97%E2%88%AB%5B%EF%BC%882x%2B1%29%2F%28x%5E2%2B2x%2B2%5D%EF%BC%89dx)
高数简单微分计算∫[(2x+1)/(x^2+2x+2])dx
高数简单微分计算∫[(2x+1)/(x^2+2x+2])dx
高数简单微分计算∫[(2x+1)/(x^2+2x+2])dx
∫(2x+1)dx/(x^2+2x+2)
=∫d(x^2+2x+2)/(x^2+2x+2)-∫dx/(x^2+2x+2)
=ln(x^2+2x+2)-arctan(x+1)+C
(2x + 1) / (x^2 + 2x + 2) = (2(x + 1) - 1) / ((x + 1)^2 + 1) = 2(x+1) / ((x + 1)^2 + 1) - 1 / ((x + 1)^2 + 1)
∫ [(2x+1)/(x^2+2x+2])dx = 2∫[ (x+1) / ((x + 1)^2 + 1)] dx- ∫ [1 / ((x + 1)^2 + 1)] ...
全部展开
(2x + 1) / (x^2 + 2x + 2) = (2(x + 1) - 1) / ((x + 1)^2 + 1) = 2(x+1) / ((x + 1)^2 + 1) - 1 / ((x + 1)^2 + 1)
∫ [(2x+1)/(x^2+2x+2])dx = 2∫[ (x+1) / ((x + 1)^2 + 1)] dx- ∫ [1 / ((x + 1)^2 + 1)] dx
= ∫ [ 1 / ((x + 1)^2 + 1)] d[(x+1)^2 + 1] - ∫ [1 / ((x + 1)^2 + 1)] d(x+1)
= 2ln((x + 1)^2 + 1) - arctan(x+1) + C
= 2ln(x^2 + 2x + 2) - arctan(x+1) + C
C为任意常数.
用到了, ∫ 1 / (1 + t^2) dt = arctan t + C.
望采纳~~
收起