Evaluate: `int_1^(e^6)[(logx)/3]dx ,w h e r e[dot]` denotes the greatest integer function.

Answer

`(e^(6)-e^(3))`

Explanation

When `1ltxlte^(3),[(logx)/3]=0`
and when `e^(3)ltxlte^(6),[(logx)/3]=1`
`:. int_(1)^(e^(6))[(logx)/3]dx=int_(1)^(e^(3))[(logx)/3]dx+int_(e^(3))^(e^(6))[(logx)/3]dx`
`=int_(1)^(e^(3)) 0dx+int_(e^(3))^(e^(6))1dx=(e^(6)-e^(3))`