The area bounded by the curve `y=x(1-log_(e)x)` and x-axis is

Options

A. `(e^(2))/(4)`
B. `(e^(2))/(2)`
C. `(e^(2)-e)/(2)`
D. `(e^(2)-e)/(4)`

Answer

A

Explanation

`y=x(1-log_(e)x)`
It meets x-axis, if `x(1-loge_(e)x)=0`
`therefore" "x=0 or x=e`
`therefore" Required area "=overset(e)underset(0)intx(1-log_(e)x)dx`
`=overset(e)underset(0)intxdx -overset(e)underset(0)intx log x dx`
`=[(x^(2))/(2)]_(0)^(e)-[(x^(2))/(2)log_(e)x]_(0)^(e)+overset(e)underset(0)int(x)/(2)dx`
`=(e^(2))/(2)-[(e^(2))/(2)-underset(xrarr0)lim(x^(2))/(2)log_(e)x]+(e^(2))/(4)`
`=(e^(2))/(4)`