∫(2,1)lnxdx与∫(2,1)(lnx)^3dx比较大小
问题描述：
∫(2,1)lnxdx与∫(2,1)(lnx)^3dx比较大小
答
∫(2,1)lnxdx
=[xlnx](2,1) - ∫(2,1) dx
= 2ln2 - (2-1)
=2ln2-1
=0.3863
∫(2,1)(lnx)^3dx
=[x(lnx)^3](2,1) - 3∫(2,1)(lnx)^2dx
=2(ln2)^3-3[x(lnx)^2](2,1) + 6∫(2,1)(lnx)dx
=2(ln2)^3 - 6(ln2)^2+ 6[xlnx](2,1)-6∫(2,1)dx
=2(ln2)^3 - 6(ln2)^2+12ln2- 6
=0.1011
ie ∫(2,1)lnxdx >∫(2,1)(lnx)^3dx
"Or"
for (1,2)
lnx > (lnx)^3
=> ∫(2,1)lnxdx > ∫(2,1)(lnx)^3dx