求不定积分 得解题思路及过程
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:2024/06/30 22:11:03
求不定积分 得解题思路及过程
∫[√﹙x+1﹚-1]/[√﹙x+1﹚+1]dx
∫[√﹙x+1﹚-1]/[√﹙x+1﹚+1]dx
![求不定积分 得解题思路及过程](/uploads/image/z/3538209-57-9.jpg?t=%E6%B1%82%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86+%E5%BE%97%E8%A7%A3%E9%A2%98%E6%80%9D%E8%B7%AF%E5%8F%8A%E8%BF%87%E7%A8%8B)
令√(x+1)=t,则x=t^2-1
所以原式=∫(t-1)/(t+1)*2tdt
=2∫(t^2-t)/(t+1)dt
=2∫(t^2+t-2t-2+2)/(t+1)dt
=2∫tdt-4∫dt+4∫d(t+1)/(t+1)
=t^2-4t+4ln|t+1|+C
=x+1-4√(x+1)+4ln(√(x+1)+1)+C
=x-4√(x+1)+4ln(√(x+1)+1)+C
所以原式=∫(t-1)/(t+1)*2tdt
=2∫(t^2-t)/(t+1)dt
=2∫(t^2+t-2t-2+2)/(t+1)dt
=2∫tdt-4∫dt+4∫d(t+1)/(t+1)
=t^2-4t+4ln|t+1|+C
=x+1-4√(x+1)+4ln(√(x+1)+1)+C
=x-4√(x+1)+4ln(√(x+1)+1)+C