作业帮问两个不定积分问题?
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/05 11:34:30
问两个不定积分问题?
问两个不定积分问题?
问两个不定积分问题?
1.∫(1-lnx)/(x-lnx)^2dx
=∫[(x-lnx)*(1/x)-(1-1/x)lnx]/(x-lnx)^2 dx
=∫[(lnx)'(x-lnx)-(x-lnx)'lnx]/(x-lnx)^2 dx
=∫[lnx/(x-lnx)]'dx
=lnx/(x-lnx)+C1
=lnx/(x-lnx)+1+C
=x/(x-lnx)+C
直接拆分也可得
∫(1-lnx)/(x-lnx)^2dx
=∫(x-x+1-lnx)/(x-lnx)^2dx
=∫(x-lnx)/(x-lnx)^2dx+∫(1-x)/(x-lnx)^2dx
=∫1/(x-lnx)dx+∫xd[1/(x-lnx)]
=∫1/(x-lnx)dx+x/(x-lnx)-∫1/(x-lnx)dx
=x/(x-lnx)+C
2.∫(x^2+1)/(x^4+1)dx
=∫[(1+1/x^2)/(x^2+1/x^2)]dx
=∫{1/[(x-1/x)^2+2]}d(x-1/x)
=∫[1/(t^2+2)]dt
=∫(√2/2)[1/(t/√2)^2+1]d(t/√2)
=(√2/2)arctan(t/√2)+C
=(√2/2)arctan[(x - 1/x)/√2]+C
C为任意常数
问题呢？